LU Factorizations for ℕ × ℕ-Matrices and Solutions of the k[S]-Hierarchy and Its Strict Version

Abstract

Let S be the $\mathbb{N}\times \mathbb{N}$-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra $k\left[S\right]$, together with the evolution equations of the deformations of the powers $\{S^i,i⩾1\}$. They are called the $k\left[S\right]$-hierarchy and the strict $k\left[S\right]$-hierarchy. For suitable Banach spaces B, we explain how LU factorizations in $GL\left(B\right)$ can be used to produce dressing matrices of both hierarchies. These dressing matrices correspond to bounded operators on B, a class far more general than the one used at a prior construction. This wider class of solutions of both hierarchies makes it possible to treat reductions of both systems. The matrix coefficients of these matrices are shown to be quotients of analytic functions. Moreover, we present a subgroup $G_{cpt}\left(B\right)$ of $GL\left(B\right)$ such that the procedure with LU factorizations works for each $g\in G_{cpt}\left(B\right)$.

1. Introduction

To make the introduction as accessible as possible to the reader, we put the topic of the present paper in a historical context, thereby highlighting the results relevant for its content. The study of integrable or exactly solvable systems has a long history. Classical mathematicians such as Euler, Lagrange, Liouville, Riemann and Poincaré amongst many others, investigated nonlinear systems which could be integrated more or less explicitly. We start at the beginning of the area of soliton theory of which we give a short description based on [1]. The first recorded observation of a solitary wave is the one made by Russell in 1834 [2]. He observed along a narrow channel in Scotland that, when a barge was suddenly stopped, the water mass put in motion by the boat formed at the prow of the barge a solitary wave that moved forward for quite some time at a constant speed and maintaining its height. After returning from this trip, he started careful experiments with water waves to classify them and after some time he came empirically to a number of results: the existence of solitary waves, a formula for their speed and a nonlinear differential equation they had to satisfy. The astronomer Airy, based on his own work on water waves, denied the existence of solitary waves, and this led to bitter disputes. This controversy around the validity of Russell’s claims went on till 1895, when Korteweg and his student de Vries published the paper [3] on propagation of shallow water waves in a narrow rectangular canal. It settled the controversy entirely in favor of Russell. In their paper, they argued that the height $u=u(x,t)$ of the waves in the canal should satisfy the differential equation

$${\displaystyle \frac{\partial u}{\partial t}}={\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}+{\displaystyle \frac{3}{2}}u{\displaystyle \frac{\partial u}{\partial x}},$$

(1)

where x is a space coordinate along the canal and t a time coordinate. Moreover, they gave a set of solitary wave solutions of this equation, the so-called cnoidal waves. Since then, Equation (1) has been called the KdV equation.

Interest in the KdV equation reignited when Zabusky and Kruskal [4] found remarkable interaction properties for the solitary wave solutions to KdV. They coined the name solitons for these solutions. Soon after that, Gardner, Greene, Kruskal and Miura [5] introduced inverse scattering transform or IST to solve the Cauchy problem for KdV, which turned out to be applicable to many more equations. As a result, the research on both theoretical and practical aspects of these equations intensified. E.g., in 1995, at the centennial of the eponymous KdV paper, the practical applications [6] of the KdV equation alone were recognized in various fields: fluid mechanics, optics, oceanography, plasma physics, astronomy and electrical transmission lines, among others.

An alternative formulation of the KdV equation was found by Lax [7]. He considered the Schrödinger operator $L_2={\partial }^2+u$ and the third order operator $A={\partial }^3+3u\partial +{\displaystyle \frac{3}{2}}\partial \left(u\right)$, where u is a function depending on x, t and possibly other parameters and ∂ denotes the partial derivative ${\displaystyle \frac{\partial }{\partial x}}$. Then, a direct calculation yields that their commutator is a zero-th order operator in ∂

$$[A,L_2]={\displaystyle \frac{1}{4}}{\partial }^3\left(u\right)+{\displaystyle \frac{3}{2}}\partial \left(u\right)u$$

and we see that the evolution equation for $L_2$

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}\left(L_2\right)=[A,L_2].\end{array}$$

(2)

is equivalent to u satisfying the KdV equation. Equation (2) was soon called the Lax form of the KdV. This form suggests that $L_2$ is obtained by conjugating a t-independent operator like ${\partial }^2$ with a t-dependent one and we will see it coming back at all kinds of integrable equations. For example, the Lax form of the nonlinear Schrödinger equation was found by Zacharov and Shabat in [8] and NLS was the starting point in [9] to treat Hamiltonian methods for soliton equations. It laid the foundation for the quantum version of IST developed by the Fadeev school in St. Petersburg, see [10]. This also brought solvable models from statistical mechanics into the picture and likewise quantum groups made their appearance in the integrable world. A rich collection of contributions to these topics and integrability can be found in [11].

The third nonlinear equation that needs to be mentioned is a two-dimensional version of KdV that was used in plasma physics by Khadomtsev and Petviashvilii and is therefore called the KP equation. For a function $u=u(x,y,t)$, it reads

$${\displaystyle \frac{3}{4}}{\displaystyle \frac{{\partial }^{2}u}{\partial y^2}}={\displaystyle \frac{\partial }{\partial x}}\left({\displaystyle \frac{\partial u}{\partial t}}-{\displaystyle \frac{1}{4}}{\displaystyle \frac{{\partial }^{3}u}{\partial x^3}}-{\displaystyle \frac{3}{2}}u{\displaystyle \frac{\partial u}{\partial x}}\right).$$

(3)

Let $v=v(x,y,t)$ be another function, $B_2$ the operator ${\partial }^2+u$ and $B_3$ the operator ${\partial }^3+{\displaystyle \frac{3}{2}}u\partial +v$. Consider the following linear system for $\psi =\psi (x,y,t)$:

$${\displaystyle \frac{\partial \psi }{\partial y}}=B_2\psi \mathrm{and}{\displaystyle \frac{\partial \psi }{\partial t}}=B_3\psi .$$

(4)

Then, the compatibility of this system is ${\displaystyle \frac{\partial B_2}{\partial t}}-{\displaystyle \frac{\partial B_3}{\partial y}}+[B_2,B_3]=0$. Eliminating v from these equations gives you that u satisfies the KP Equation (3). Krichever searched for an analogue for the KP equation of the results in [12,13,14,15] for the KdV equation and succeeded [16,17]. The algebraic geometric data he needed were the following: a complete irreducible algebraic curve X, a rank 1 torsion free coherent sheaf $\mathcal{L}$ over X, a non-singular point $x_{\infty }$, a local coordinate ${\displaystyle \frac{1}{z}}$ around $x_{\infty }$ such that z is an isomorphism between a closed neighbourhood $X_{\infty }$ of $x_{\infty }$ and the unit disc around infinity on the Riemann sphere and finally a trivialization $\phi$ of $\mathcal{L}$ over $X_{\infty }$. If X is nonsingular, then X is a compact Riemann surface and $\mathcal{L}$ is a line bundle over X.

The fourth type of system that exhibit soliton-like behavior was found by M.Toda [18]. He was inspired by results of Fermi [19] and Ford [20] and his goal was to elucidate certain characteristic features of nonlinear waves, using nonlinear lattice models. He found lattice models that possess multi-soliton solutions, periodic waves and solvability of the initial value problem with IST. Since then, they carry his name. We mention one of them, the finite Toda lattice. For more examples, we refer the reader to [21]. The finite Toda lattice describes the motion of n particles, all with mass equal to 1, on a line with only nearest neighbor interaction. Let $q_j,j=1,\cdots ,n$ denote the position of the j-th particle and $p_j={\displaystyle \frac{d}{dt}}\left(q_j\right)$ its momentum. The equations of motion of the n particles are:

$${\displaystyle \frac{d^2\left(q_j\right)}{d{t}^2}}=e^{q_{j-1}-q_j}-e^{q_j-q_{j+1}},$$

(5)

where we put $e^{q_0}=0$ and $e^{-q_{n+1}}=0$. Introduce the new coordinates

$$a_j={\displaystyle \frac{1}{2}}{e}^{{\displaystyle \frac{1}{2}}(q_j-q_{j+1})},1⩽j⩽n-1,\mathrm{and}{b}_j=-{\displaystyle \frac{1}{2}}{p}_j,1⩽j⩽n,$$

and recall that $a_0=0$. Then, the equations of motion (5) transform into

$${\displaystyle \frac{d\left(a_j\right)}{dt}}=a_j(b_{j+1}-b_j)\mathrm{and}{\displaystyle \frac{d\left(b_j\right)}{dt}}=b_j(a_j^2-a_{j-1}^2).$$

(6)

It was H. Flaschka [22] who showed that Equation (6) can be written in Lax form. Consider the following $n\times n$-matrices

$$L=\left(\begin{array}{ccccc}{b}_1& a_1& 0& \cdots & 0\\ a_1& b_2& a_2& \ddots & ⋮\\ 0& a_2& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & a_{n-1}\\ 0& 0& \cdots & a_{n-1}& b_n\end{array}\right)\mathrm{and}B=\left(\begin{array}{ccccc}0& a_1& 0& \cdots & 0\\ -a_1& 0& a_2& \ddots & ⋮\\ 0& -a_2& \ddots & \ddots & 0\\ ⋮& \ddots & \ddots & \ddots & a_{n-1}\\ 0& 0& \cdots & -a_{n-1}& 0\end{array}\right).$$

Then, one verifies directly that Equation (6) amounts to the Lax equation

$${\displaystyle \frac{dL}{dt}}=BL-LB=[B,L].$$

(7)

With the help of this equation, one shows that the eigenvalues of L are time-independent and polynomials in $\left\{b_j\right\}$ and $\left\{a_j^2\right\}$. Moreover, the functions

$$F_k(q,p)=\mathrm{Trace}\left(L^k\right),k=1,\cdots ,n,$$

are n linear independent integrals for the vector field

$$X_H=\sum _{j=1}^n\left({\displaystyle \frac{\partial H}{\partial p_j}}{\displaystyle \frac{\partial }{\partial q_j}}-{\displaystyle \frac{\partial H}{\partial q_j}}{\displaystyle \frac{\partial }{\partial p_j}}\right)$$

for the Hamiltonian $H(q,p)={\displaystyle \frac{1}{2}}{\sum }_{j=1}^n{p}_j^2+{\sum }_{j=1}^{n-1}{e}^{q_j-q_{j+1}}$ and $\left\{F_k\right\}$ are in involution with respect to the standard Poisson bracket $\{·,·\}$ on ${\mathbb{R}}^{2n}$, i.e., $\{F_{k_1},F_{k_2}\}=0$. For proofs, we refer the reader to Chapter 3 in [23]. Note that the matrix B in the Lax form of the finite Toda lattice is the anti-symmetric part of L in the splitting of the $n\times n$-matrices in an anti-symmetric part and a lower triangular part. Hence, (7) is a Lax equation coupled to this decomposition. An integrable hierarchy, coupled to an infinite dimensional variant of this decomposition, is described in [24]. This decomposition is different from the one on which the discrete KP hierarchy [25] is based.

Gelfand and Dickey realized that the third order operator A occurring in the Lax form (2) of KdV was nothing but the differential operator part of the third power of the square root ${\left(L_2\right)}^{{\displaystyle \frac{1}{2}}}$ of $L_2$ in the pseudo-differential operators Psd in ∂. Here, Psd is an extension of the differential operators with coefficients, functions on which ∂ acts via Leibniz. It consists of possibly infinite series ${\sum }_{i⩽N}{p}_i{\partial }^i$ satisfying $\partial {\partial }^{-1}=1$ and ${\partial }^{-1}p={\sum }_{k=0}^{\infty }{(-1)}^k{\partial }^k\left(p\right){\partial }^{-1-k}$ and it has the property that each monic element of Psd of order $m⩾1$ possesses an m-th root in Psd. Then, it was clear how to generalize it. In [26], the authors considered, instead of the Schrödinger operator $L_2$, a monic differential operator $L_n$ of order $n,n⩾2,$ in ∂ of the form $L_n={\partial }^n+{\sum }_{i=0}^{n-2}{u}_i{\partial }^i$ and for all $m⩾1$ the monic differential operator $A_m$ consisting of the differential operator part of the m-th power of the n-th root ${\left(L_n\right)}^{{\displaystyle \frac{1}{n}}}$ of $L_n$ in the pseudo-differential operators. Take the infinite set of Lax equations for $L_n$

$${\displaystyle \frac{\partial }{\partial t_m}}\left(L_n\right)=[A_m,L_n],m⩾1.$$

(8)

Gelfand and Dickey showed that this system of equations is compatible and determined its Hamiltonian structure. For $n=2$, this system is usually called the KdV hierarchy and for the generalization one uses the name n-th Gelfand–Dickey hierarchy or n-KdV hierarchy. The strict version of the n-th Gelfand–Dickey hierarchy is a wider deformation of ${\partial }^n$ and is treated in [27].

A next big step was the appearance of a long series of papers by members of the Sato school in Kyoto on transformation groups for soliton equations. They corresponded with various infinite dimensional groups and were full of all kinds of intriguing ideas and connections. We focus on the main example: the KP hierarchy, that is the one asssociated with the group $GL(\infty )$. Date, Jimbo, Kashiwara and Miwa introduced it in [28] and considered elements L in Psd of the form $L=\partial +{\sum }_{j=1}{\ell }_{j+1}{\partial }^{-j}$ and we see $\mathbb{C}\left[L\right]$ as a deformation of the commutative algebra $\mathbb{C}[\partial ]$. Note that any $L=K\partial K^{-1}$ is of this form. In that case, we are dealing with a deformation of $\mathbb{C}[\partial ]$ by conjugating with the dressing operator K. We assume that the coefficients in the pseudo-differential operators are differentiable with respect to the parameters $\{t_k\mid k⩾1\}$ and that each ${\displaystyle \frac{\partial }{\partial t_k}}$ commutes with ∂. Let $B_k,k⩾1,$ be the differential operator part of $L^k$, then the evolution equations for L are

$${\displaystyle \frac{\partial }{\partial t_k}}\left(L\right)=[B_k,L],k⩾1.$$

(9)

These Lax equations are equivalent to the zero curvature relations for $\left\{B_k\right\}$

$${\displaystyle \frac{\partial }{\partial t_{k_1}}}\left(B_{k_2}\right)-{\displaystyle \frac{\partial }{\partial t_{k_2}}}\left(B_{k_1}\right)=[B_{k_1},B_{k_2}].$$

(10)

Now, the first two nontrivial powers of L start as follows

$$\begin{array}{c}\hfill L^2={\partial }^2+2{\ell }_2+(2{\ell }_3+\partial \left({\ell }_2\right)){\partial }^{-1}+\cdots ,L^3={\partial }^3+3{\ell }_2\partial +(3{\ell }_3+3\partial \left({\ell }_2\right))+\cdots \end{array}$$

Hence, $B_2={\partial }^2+2{\ell }_2$ and $B_3={\partial }^3+3{\ell }_2\partial +(3{\ell }_3+3\partial \left({\ell }_2\right))$ and from the calculation at the introduction of the KP equation it follows that $2{\ell }_2$ is a solution of the KP equation. This justifies the name of the hierarchy. Each of the Gelfand–Dickey hierarchies is a subsystem of the KP hierarchy. Any solution L of the KP hierarchy for which $L^n$ is a differential operator delivers the solution $L^n$ of the n-th Gelfand–Dickey and in terms of deformations this corresponds to deforming the commutative algebra $\mathbb{C}\left[{\partial }^n\right]$ inside the differential operators. The geometric picture sketched in [28] is that the group $GL(\infty )$ acts in the unspecified space of $\tau$-functions, that the $GL(\infty )$-orbit of the vacuum is the Sato Grassmannian and this $GL(\infty )$-orbit is the solutions of the KP hierarchy. This could use some clarification. In a later paper, the authors specified the space of $\tau$-functions as the polynomials in $\left\{t_k\right\}$ and the picture became better, but not good enough for, e.g., the solutions found by Krichever. Segal and Wilson followed a different path in [29] to obtain solutions of the KP hierarchy. They used the linearization of KP. This is a set of relations in the Psd-module of oscillating functions

$$L\psi =z\psi ,{\displaystyle \frac{\partial }{\partial t_k}}\left(\psi \right)=B_k\left(\psi \right),k⩾1.$$

(11)

If you find for L and all the $B_k$ a suitable $\psi$ in that module such that these relations hold, then L is a solution of the KP hierarchy. Segal and Wilson introduced a Grasmann manifold based on a space of Hilbert–Schmidt operators from which they constructed solutions for the KP hierarchy and for each point of this variety they introduced a $\tau$-function that relates to the wave function in the way, the Sato school predicted. The solutions found by Krichever fall also within this framework. The strict KP hierarchy is a wider deformation of $\mathbb{C}[\partial ]$ introduced in [30]. The geometric picture is a flag variety covering the Grassmann manifold of Segal and Wilson and can be found in [31]. In this case, one needs two $\tau$-functions to describe the relevant wave function [32]. It will not come as a surprise that the strict versions of the Gelfand–Dickey hierarchies are subsystems of the strict KP hierarchy.

We have seen how research in integrable systems has developed from single equations to infinite dimensional integrable hierarchies. They play an important role in both mathematics and theoretical physics and common grounds are a big stimulus for cooperation and success. We mention a few areas in physics where they play an important role. First of all, they often occur in matrix models at the application of large N-techniques, see, e.g., Refs. [33,34,35]. The next subject combines a wide range of disciplines: two-dimensional quantum field theory, intersection theory on moduli spaces of Riemann surfaces, integrable hierarchies, matrix integrals and random surfaces to name a few. At the beginning of the 1990s, Edward Witten [36] started a program [37] to search for a relationship between random surfaces and the algebraic topology of moduli space and formulated a series of conjectures based on his findings. The first of these conjectures was proved by Kontsevich [38], who showed that the string partition function is a $\tau$-function of the KdV hierarchy. Another topic is topological strings on Calabi–Yau geometries. It provides a unifying picture connecting non-critical (super)strings, integrable hierarchies and various matrix models, see [39,40].

In [41], we studied integrable hierarchies related to deforming various commutative subalgebras of the $\mathbb{N}\times \mathbb{N}$-matrices with coefficients in k, $k=\mathbb{R}$ or $k=\mathbb{C}$. We also constructed solutions of these hierarchies by conjugating the original matrices with matrices from operators of the form identity plus a Hilbert–Schmidt operator. In operator terms, this is a mild perturbation as Hilbert–Schmidt operators are compact. In a later paper [42], we took the main example of a commutative subalgebra and studied the scaling properties of the corresponding hierarchy. Here, we stick to the basic example and present a far wider class of dressing matrices for both hierarchies with matrices corresponding to bounded operators on suitable Banach spaces. To be more concrete, we first need some notation.

Let R be a commutative k-algebra over the field k. We write $M_n\left(R\right)$ for the $n\times n$-matrices with coefficients from R and similarly $M_{\mathbb{N}}\left(R\right)$ for the space of $\mathbb{N}\times \mathbb{N}$-matrices with coefficients from R. The transpose $A^T$ of any matrix $A\in M_{\mathbb{N}}\left(R\right)$ or any $1\times \mathbb{N}$ matrix A with coefficients from R is defined as in the finite dimensional case. Let V be the R-module of all $\mathbb{N}\times 1$-matrices with coefficients from R, i.e.,

$$V=R^{\mathbb{N}}=\{\overrightarrow{x}=\left(x_j\right)={\left(\begin{array}{cccc}{x}_0& x_1& x_2& \cdots \end{array}\right)}^T\mid x_j\in R\mathrm{for}\mathrm{all}j\in \mathbb{N}\}.$$

Inside V, we consider the R-submodule

$$V_{\mathrm{fin}}=\{\overrightarrow{x}=\left(x_j\right)\in V\mid x_j\ne 0\mathrm{for}\mathrm{a}\mathrm{finite}\mathrm{number}\mathrm{of}j\}$$

The space $V_{\mathrm{fin}}$ is a free R-module with $\{\overrightarrow{e}\left(i\right)\mid i\in \mathbb{N}\}$ as the basis, where $\overrightarrow{e}\left(i\right)$ is the vector with the i-th coordinate equal to one and the remaining ones equal to zero. For each $\overrightarrow{x}\in V_{\mathrm{fin}}$ and each $A\in M_{\mathbb{N}}\left(R\right)$, the product $A\overrightarrow{x}$ is well defined and determines a vector in V. Hence, if we write

$$M_A\left(\overrightarrow{x}\right):=A\overrightarrow{x},$$

then $M_A$ is an R-linear map in ${\mathrm{Hom}}_R(V_{\mathrm{fin}},V)$. Inside $M_{\mathbb{N}}\left(R\right)$, we have the basic matrices $E_{(i,j)},i$ and $j\in \mathbb{N}$, whose matrix entries, in Kronecker notation, are given by

$${\left(E_{(i,j)}\right)}_{mn}={\delta }_{im}{\delta }_{jn}.$$

It is convenient to use the notation $A={\sum }_{i,j}{a}_{(i,j)}{E}_{(i,j)}$ for an $A=\left(a_{(i,j)}\right)\in L{T}_{\mathbb{N}}\left(R\right)$. A central role in this paper is played by the shift matrix S, its transpose $S^T$ and their powers, where S is the matrix ${\sum }_{s\in \mathbb{N}}{E}_{(s,s+1)}$ that corresponds to the shift operator $M_S:V\to V$ defined by

$$M_S\left({\left(\begin{array}{cccc}{x}_0& x_1& x_2& \cdots \end{array}\right)}^T\right)={\left(\begin{array}{cccc}{x}_1& x_2& x_3& \cdots \end{array}\right)}^T.$$

(12)

The way the two deformations of the commutative algebra $k\left[S\right]$ are performed is by conjugating them with invertible lower triangular $\mathbb{N}\times \mathbb{N}$-matrices with coefficients from R. At one deformation, we conjugate with lower triangular matrices with the identity on the central diagonal. The evolution equations of the deformed generators are Lax equations and are determined by the decomposition of a matrix in $M_{\mathbb{N}}\left(R\right)$ in their upper triangular and strict lower triangular parts. The evolution equations of the deformed generators in this case are called the Lax equations of the $k\left[S\right]$-hierarchy.

For the second wider deformation, the central diagonal is merely assumed to be invertible. Also in this case, the evolution equations of the deformed generators are Lax equations. Only now they are based on the splitting of $M_{\mathbb{N}}\left(R\right)$ matrices in their strict upper triangular and lower triangular parts. The evolution equations of these deformations of $\{S^i\mid i⩾1\}$ are called the Lax equations of the strict $k\left[S\right]$-hierarchy.

In the case of the $k\left[S\right]$-hierarchy, the dressing matrix of the deformation turns out to be unique, and in the strict case it is determined up to a multiple of the identity. The uniqueness of the dressing operator enables one to prove directly the equivalence of the Lax form of the k[S]-hierarchy with a set of Sato–Wilson equations. There exists an analogue of the Sato–Wilson equations for the strict $k\left[S\right]$-hierarchy. It always implies the Lax equations of this hierarchy and it suffices for the geometric solutions that we produce further on. However, we show how they can fail.

Solutions of both hierarchies are constructed by producing wave matrices in the linearization module of each hierarchy. Therefore, we recall the essentials of this approach. We start the last section with the description of a family of Banach spaces B that all have a countable Schauder basis; they have the approximation property and the elements with only a finite number of coordinates with respect to the Schauder basis are dense in B. For these spaces, we show how LU factorizations in $GL\left(B\right)$ can be used to construct for a nonempty open subset of $GL\left(B\right)$ solutions of the $k\left[S\right]$-hierarchy and its strict version. We show that the dressing matrix coefficients of these solutions are quotients of analytic functions on the group of commuting flows of the hierarchies. Next, we introduce a subgroup $G_{cpt}\left(B\right)$ of $GL\left(B\right)$ that is the analogue of the restricted linear group ${GL}_{res}\left(B\right)$ that was used in cases in which B is a Hilbert space to construct solutions of the KP hierarchy. We conclude by showing that $G_{cpt}\left(B\right)$ is contained in this open subset and that leads to a description of each set of solutions in terms of a homogeneous space of $G_{cpt}\left(B\right)$.

The content of the various sections is as follows: Section 2 describes the scene of the deformations, the algebra $L{T}_{\mathbb{N}}\left(R\right)$, the maximality of $k\left[S\right]$ and the properties required later on. The next section is devoted to a description of the two deformations; it contains a discussion of the Lax equations they have to satisfy and we describe there the link with their Sato–Wilson form. The form of the relevant $L{T}_{\mathbb{N}}\left(R\right)$-module, the equations of the linearization and a characterization of the special vectors, called wave matrices, can all be found in Section 4. In the last section, we present the role of LU factorization in constructing solutions of both hierarchies, we discuss properties of the dressing matrix coefficients and prove that $G_{cpt}\left(B\right)$ is contained in the open set of $GL\left(B\right)$, where LU factorization is effective in producing solutions of the hierarchies.

2. The Algebra ${\mathit{LT}}_{\mathbb{N}}\left(\mathit{R}\right)$

The algebra $L{T}_{\mathbb{N}}\left(R\right)$ is built with the powers of S discussed above and its transpose $S^T$, plus the diagonal matrices. Let $\left\{d\right(s)\mid s\in \mathbb{N}\}$ be a set of elements in R. Then, the diagonal matrix diag($d\left(s\right)$) in $M_{\mathbb{N}}\left(R\right)$ is given by

$$\mathrm{diag}\left(d\left(s\right)\right)=\mathrm{diag}(d\left(0\right),d\left(1\right),\cdots ):=\sum _{s\in \mathbb{N}}d\left(s\right)E_{(s,s)}=\left(\begin{array}{cccc}d\left(0\right)& 0& 0& \cdots \\ 0& d\left(1\right)& 0& \ddots \\ 0& 0& d\left(2\right)& \ddots \\ ⋮& \ddots & \ddots & \ddots \end{array}\right).$$

The algebra of all diagonal matrices in $M_{\mathbb{N}}\left(R\right)$ is denoted by

$${\mathcal{D}}_{\mathbb{N}}\left(R\right)=\{d=\mathrm{diag}\left(d\left(s\right)\right)|d\left(s\right)\in R\mathrm{for}\mathrm{all}s\in \mathbb{N}\}.$$

One has a diagonal embedding $i_1$ from R into ${\mathcal{D}}_{\mathbb{N}}\left(R\right)$ by taking all diagonal coefficients of $i_1\left(r\right)$ equal to r, i.e.,

$$i_1\left(r\right)=\left(\begin{array}{cccc}r& 0& 0& \cdots \\ 0& r& 0& \ddots \\ 0& 0& r& \ddots \\ ⋮& \ddots & \ddots & \ddots \end{array}\right).$$

Next, we decompose a matrix $A=\left(a_{ij}\right)\in M_{\mathbb{N}}\left(R\right)$ into its diagonals. For $m⩾0$, the m-th diagonal of A is by definition the matrix

$$d_m\left(A\right)S^m=\mathrm{diag}\left(a_{(s,s+m)}\right)S^m=\sum _{i⩾0}{a}_{(i,i+m)}{E}_{(i,i+m)}$$

and those diagonals are called positive. Similarly, for $m⩽0,$ the m-th diagonal of A is defined as the negative matrix

$${\left(S^T\right)}^{-m}{d}_m\left(A\right)={\left(S^T\right)}^{-m}\mathrm{diag}\left(a_{(s-m,s)}\right)=\sum _{i⩾0}{a}_{(i-m,i)}{E}_{(i-m,i)}.$$

So each matrix $A\in M_{\mathbb{N}}\left(R\right)$ decomposes uniquely as

$$A=\sum _{m⩾0}{d}_m\left(A\right)S^m+\sum _{m<0}{\left(S^T\right)}^{-m}{d}_m\left(A\right).$$

(13)

Definition 1.

Let $L{T}_{\mathbb{N}}\left(R\right)$ be the collection of all matrices in $M_{\mathbb{N}}\left(R\right)$ that have only a finite number of nonzero positive diagonals.

Note that the multiplication between S and its transpose is not commutative

$$S{S}^T=Id\mathrm{and}{S}^{T}S=\sum _{i⩾1}{E}_{(i,i)}.$$

(14)

Since the interaction between the diagonal matrices and the powers of S and $S^T$ is determined by the relations

$$\begin{array}{c}\hfill S\mathrm{diag}\left(d\right(0),d(1),\cdots )=\mathrm{diag}\left(d\right(1),d(2),\cdots )S\mathrm{and}\end{array}$$

(15)

$$\begin{array}{c}\hfill \mathrm{diag}(d\left(0\right),d\left(1\right),\cdots )S^T=S^T\mathrm{diag}(d\left(1\right),d\left(2\right),\cdots ),\end{array}$$

(16)

it follows from relations (14), (15) and (16) that the collection $L{T}_{\mathbb{N}}\left(R\right)$ is an algebra with respect to matrix multiplication. We use the decomposition (13) to assign a degree to elements of $L{T}_{\mathbb{N}}\left(R\right)$. For a nonzero $A\in L{T}_{\mathbb{N}}\left(R\right)$, the degree is equal to m if its highest nonzero diagonal is the m-th and the degree of the zero element is $-\infty$.

Lemma 1

([42]). The centralizer in $L{T}_{\mathbb{N}}\left(R\right)$ of the matrix S consists of

$$\{\sum _{j⩾0}{i}_1\left(r_j\right)S^j\mid r_j\in R\}.$$

A consequence of Lemma 1 is the following property of $k\left[S\right]=\{{\sum }_{i=0}^N{k}_i{S}^i\mid k_i\in k\}$:

Corollary 1.

The algebra $k\left[S\right]$ is a maximal commutative subalgebra of $L{T}_{\mathbb{N}}\left(k\right)$.

As any associative algebra, $L{T}_{\mathbb{N}}\left(R\right)$ is a Lie algebra with the commutator as a bracket. We use two decompositions of $L{T}_{\mathbb{N}}\left(R\right)$ into the direct sum of two Lie subalgebras. The first splits elements of $L{T}_{\mathbb{N}}\left(R\right)$ as follows

$$A={\pi }_{ut}\left(A\right)+{\pi }_{slt}\left(A\right)=\sum _{m⩾0}{d}_m\left(A\right)S^m+\sum _{m<0}{\left(S^T\right)}^m{d}_m\left(A\right)$$

and the second as

$$A={\pi }_{sut}\left(A\right)+{\pi }_{lt}\left(A\right)=\sum _{m>0}{d}_m\left(A\right)S^m+\sum _{m⩽0}{\left(S^T\right)}^m{d}_m\left(A\right).$$

The first way to split elements of $L{T}_{\mathbb{N}}\left(R\right)$ yields the Lie algebra decomposition

$$\begin{array}{cc}\hfill & L{T}_{\mathbb{N}}\left(R\right)={\pi }_{ut}\left(L{T}_{\mathbb{N}}\left(R\right)\right)\oplus {\pi }_{slt}\left(L{T}_{\mathbb{N}}\left(R\right)\right),\mathrm{where}\hfill \\ \hfill & {\pi }_{ut}\left(L{T}_{\mathbb{N}}\left(R\right)\right)=\{A\in L{T}_{\mathbb{N}}\left(R\right)\mid {\pi }_{ut}\left(A\right)=A\},\hfill \\ \hfill & {\pi }_{slt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)=\{A\in L{T}_{\mathbb{N}}\left(R\right)\mid {\pi }_{slt}\left(A\right)=A\}.\hfill \end{array}$$

The second leads to

$$\begin{array}{cc}\hfill & L{T}_{\mathbb{N}}\left(R\right)={\pi }_{sut}\left(L{T}_{\mathbb{N}}\left(R\right)\right)\oplus {\pi }_{lt}\left(L{T}_{\mathbb{N}}\left(R\right)\right),\mathrm{where}\hfill \\ \hfill & {\pi }_{sut}\left(L{T}_{\mathbb{N}}\left(R\right)\right)=\{A\in L{T}_{\mathbb{N}}\left(R\right)\mid {\pi }_{sut}\left(A\right)=A\},\hfill \\ \hfill & {\pi }_{lt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)=\{A\in L{T}_{\mathbb{N}}\left(R\right)\mid {\pi }_{lt}\left(A\right)=A\}.\hfill \end{array}$$

Inside $L{T}_{\mathbb{N}}\left(R\right)$, we associate a group with each of the Lie subalgebras ${\pi }_{slt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)$ and ${\pi }_{lt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)$. Note that on ${\pi }_{slt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)$ the exponential map is well defined and yields elements in

$${\mathcal{U}}_{-}={\mathcal{U}}_{-}\left(R\right)=\{Id+Y\mid Y\in {\pi }_{slt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)\}.$$

One can easily verify that ${\mathcal{U}}_{-}$ is a group with respect to multiplication. We see ${\mathcal{U}}_{-}$ as the group corresponding to ${\pi }_{slt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)$. If the exponential map is well defined on ${\pi }_{lt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)$, then the resulting elements belong to the group

$${\mathcal{P}}_{-}={\mathcal{P}}_{-}\left(R\right)=\{A=\sum _{m⩽0}{\left(S^T\right)}^m{d}_m\left(A\right)\mid d_0\left(A\right)=\mathrm{diag}\left(d\left(s\right)\right),\mathit{all}d\left(s\right)\in R^{\ast }\}.$$

and therefore we see ${\mathcal{P}}_{-}$ as the group associated with ${\pi }_{lt}\left(L{T}_{\mathbb{N}}\left(R\right)\right)$.

3. The $\mathit{k}\left[\mathit{S}\right]$-Hierarchy and Its Strict Version

In this section, we discuss the two deformations of $k\left[S\right]$ that we consider and the evolution equations we want the deformations of S to satisfy. At the first deformation, each ${\sum }_{i⩾0}{k}_i{S}^i$ in $k\left[S\right]$ is deformed into ${\sum }_{i⩾0}{k}_i{\mathcal{L}}^i$, where $\mathcal{L}\in L{T}_{\mathbb{N}}\left(R\right)$ is an element of the form

$$\mathcal{L}=S+\sum _{i⩽0}{\left(S^T\right)}^i{\ell }_i,\mathrm{with}{\ell }_i\in {\mathcal{D}}_{\mathbb{N}}\left(R\right).$$

(17)

One directly checks that any element $US{U}^{-1}$, with $U\in {\mathcal{U}}_{-}$, has this form and we call $US{U}^{-1}$ a ${\mathcal{U}}_{-}$-deformation of S. We also call U the dressing matrix of $US{U}^{-1}$. At the second deformation, we transform each matrix ${\sum }_{i⩾0}{k}_i{S}^i\in k\left[S\right]$ into ${\sum }_{i⩾0}{k}_i{\mathcal{M}}^i$, where $\mathcal{M}\in L{T}_{\mathbb{N}}\left(R\right)$ is an element of the form

$$\mathcal{M}=m_{1}S+\sum _{i⩽0}{\left(S^T\right)}^i{m}_i,\mathrm{with}{m}_i\in {\mathcal{D}}_{\mathbb{N}}\left(R\right),m_1\in {\mathcal{D}}_{\mathbb{N}}{\left(R\right)}^{\ast }.$$

(18)

Also in this case, one can easily verify that any matrix $PS{P}^{-1}$, with $P\in {\mathcal{P}}_{-}$, possesses the form (18) and therefore it is called a ${\mathcal{P}}_{-}$-deformation of S. Likewise, we also call P the dressing matrix of the deformation $PS{P}^{-1}$. Moreover, we showed in [42].

Lemma 2.

Conversely, there holds for the deformations (17) and (18)

(a) 

Any $\mathcal{L}$ of the form (17) can uniquely be written in the form $\mathcal{L}=US{U}^{-1}$ with $U\in {\mathcal{U}}_{-}$, i.e., $\mathcal{L}$ is a ${\mathcal{U}}_{-}$-deformation of S.

(b) 

Any $\mathcal{M}$ of the form (18) can be written in the form $\mathcal{M}=dUS{U}^{-1}{d}^{-1}$, where $U\in {\mathcal{U}}_{-}$ is unique and $d\in {\mathcal{D}}_{\mathbb{N}}{\left(R\right)}^{\ast }$ is determined up to a factor from $i_1\left(R^{\ast }\right)$. In particular, $\mathcal{M}$ is a ${\mathcal{P}}_{-}$-deformation of S.

Next, we discuss the evolution equations that an ${\mathcal{U}}_{-}$-deformation $\mathcal{L}$ of S has to satisfy and those for a ${\mathcal{P}}_{-}$-deformation $\mathcal{M}$ of S. Thereby, each $S^i,i⩾1,$ is seen as an infinitesimal generator of a flow. In that light, we assume in both cases that R is equipped with a set of commuting k-linear derivations $\{{\partial }_i:R\to R\mid i⩾1\}$, where each ${\partial }_i$ should be seen as an algebraic substitute for the derivative with respect to the flow parameter corresponding to the flow generated by each $S^i$. By letting each ${\partial }_i$ act coefficient-wise on matrices in $L{T}_{\mathbb{N}}\left(R\right)$, we obtain a set of derivations of $L{T}_{\mathbb{N}}\left(R\right)$, also denoted by $\left\{{\partial }_i\right\}$. The data $(R,\{{\partial }_i\mid i⩾1\})$ we call a setting for both deformations.

For each ${\mathcal{U}}_{-}$-deformation $\mathcal{L}$ of S and all $i⩾1$, we consider the cut-offs

$${\mathcal{B}}_i\left(S\right):={\pi }_{ut}\left({\mathcal{L}}^i\right).$$

(19)

Note that, since all $\left\{{\mathcal{L}}^i\right\}$ commute, the $\{{\mathcal{B}}_i\left(S\right)\mid i⩾1\}$ satisfy for all $m⩾1$

$$[{\mathcal{B}}_i\left(S\right),{\mathcal{L}}^m]=-[{\pi }_{slt}\left({\mathcal{L}}^i\right),{\mathcal{L}}^m],$$

where the right-hand side is of degree $m-1$ or lower, like ${\partial }_i\left(\mathcal{L}\right)$. This shows that it makes sense to unite the following set of Lax equations for the ${\mathcal{L}}^m$ in one combined system, the so-called $k\left[S\right]$-hierarchy:

$${\partial }_i\left({\mathcal{L}}^m\right)=[{\mathcal{B}}_i\left(S\right),{\mathcal{L}}^m]=-[{\pi }_{slt}\left({\mathcal{L}}^i\right),{\mathcal{L}}^m].$$

(20)

It suffices to prove the equations just for $m=1$. For, since ${\partial }_i$ and $\mathrm{ad}\left({\mathcal{B}}_i\left(S\right)\right)$ are both k-linear derivations of $L{T}_{\mathbb{N}}\left(R\right)$, all basis elements $\{{\mathcal{L}}^m\mid m⩾1\}$ of the deformation $k\left[\mathcal{L}\right]$ of $k\left[S\right]$ satisfy the same Lax equations. Equation (20) itself is called the Lax equations of the $k\left[S\right]$-hierarchy. Note that the Lax Equation (20) shows that the action of each ${\partial }_i$ on the coefficients of $\mathcal{L}$ expresses each of them in a polynomial expression of the coefficients of $\mathcal{L}$. Any ${\mathcal{U}}_{-}$-deformation $\mathcal{L}$ of S in $L{T}_{\mathbb{N}}\left(R\right)$ that satisfies all Equation (20), $i⩾1$, is called a solution of the $k\left[S\right]$-hierarchy in the setting $(R,\left\{{\partial }_i\right\})$. Note that in each setting there is at least one solution of the $k\left[S\right]$-hierarchy, namely $\mathcal{L}=S$, the trivial solution of the $k\left[S\right]$-hierarchy. We can express the conditions when a ${\mathcal{U}}_{-}$-deformation $\mathcal{L}=US{U}^{-1}$ is a solution of the $k\left[S\right]$-hierarchy, in terms of U. For, there holds

Lemma 3.

Any $\mathcal{L}=US{U}^{-1}$ with $U\in {\mathcal{U}}_{-}$ is a solution of the $k\left[S\right]$-hierarchy if and only if U satisfies the relations: for all $i⩾1$

$${\partial }_i\left(U\right)U^{-1}=-{\pi }_{slt}\left({\mathcal{L}}^i\right).$$

(21)

Proof. 

Since ${\partial }_i\left(U^{-1}\right)=-U^{-1}{\partial }_i\left(U\right)U^{-1}$, we obtain for $\mathcal{L}=US{U}^{-1}$ that

$${\partial }_i\left(\mathcal{L}\right)=[{\partial }_i\left(U\right)U^{-1},\mathcal{L}]$$

If U satisfies (21), then $[-{\pi }_{slt}\left({\mathcal{L}}^i\right),\mathcal{L}]=[{\mathcal{B}}_i\left(S\right),\mathcal{L}]$ yields the Lax equations for $\mathcal{L}$. Conversely, if $\mathcal{L}$ is a solution of the $k\left[S\right]$-hierarchy, then ${\partial }_i\left(U\right)U^{-1}+{\pi }_{slt}\left({\mathcal{L}}^i\right)$ commutes with $\mathcal{L}$ and thus $\widehat{U}=U^{-1}({\partial }_i\left(U\right)U^{-1}+{\pi }_{slt}\left({\mathcal{L}}^i\right))U$ commutes with S. The element $\widehat{U}$ only has strict negative diagonals and Lemma 1 implies that $\widehat{U}=0$ and this proves the claim. □

Since the relations (21) are the analogue of the Sato–Wilson equations for the KP hierarchy, we call them the Sato–Wilson form of the $k\left[S\right]$-hierarchy. This is the form that comes out of the approach in Section 4 and that we will use at the construction in Section 5.

Remark 1.

Let $d_0\left({\mathcal{L}}^i\right)$ be the zero-th diagonal of ${\mathcal{L}}^i$, then the zero curvature form of the $k\left[S\right]$-hierarchy, see [42], implies that the commuting diagonal matrices $\left\{d_0\left({\mathcal{L}}^i\right)\right\}$ satisfy the compatibility conditions

$${\partial }_{i_1}\left(d_0\left({\mathcal{L}}^{i_2}\right)\right)={\partial }_{i_2}\left(d_0\left({\mathcal{L}}^{i_1}\right)\right),\mathit{for}\mathit{all}{i}_1⩾1,\mathit{and}{i}_2⩾1.$$

(22)

Next, we treat the evolution equations for the ${\mathcal{P}}_{-}$-deformations $\{{\mathcal{M}}^m\mid m⩾1\}$. In that case, we consider for each $i⩾1$ the strict cut-off

$${\mathcal{C}}_i\left(S\right):={\pi }_{sut}\left({\mathcal{M}}^i\right).$$

(23)

Since all the ${\mathcal{M}}^i$ commute, there holds

$$[{\mathcal{C}}_i\left(S\right),{\mathcal{M}}^m]=-[{\pi }_{lt}\left({\mathcal{M}}^i\right),{\mathcal{M}}^m],$$

which shows that the $\{{\mathcal{C}}_i\left(S\right)\mid i⩾1\}$ have the common property that the commutator with ${\mathcal{M}}^m$ has degree m or lower. The same holds for the matrix ${\partial }_i\left({\mathcal{M}}^m\right)$, so it makes sense to unite the following set of Lax equations for the $\left\{{\mathcal{M}}^m\right\}$ in one combined system

$${\partial }_i\left({\mathcal{M}}^m\right)=[{\mathcal{C}}_i\left(S\right),{\mathcal{M}}^m]=-[{\pi }_{lt}\left({\mathcal{M}}^i\right),{\mathcal{M}}^m].$$

(24)

Because of the form of the $\left\{{\mathcal{C}}_i\left(S\right)\right\}$ and the similarity with the Lax Equation (20), we call this system the strict $k\left[S\right]$-hierarchy. Equation (24) itself is called the Lax equations of the strict $k\left[S\right]$-hierarchy. Note that also in the strict case the Lax Equation (24) shows that the action of each ${\partial }_i$ on the coefficients of $\mathcal{M}$ expresses each of them in a polynomial expression of the matrix coefficients of $\mathcal{M}$. Any ${\mathcal{P}}_{-}$-deformation $\mathcal{M}$ of S in $L{T}_{\mathbb{N}}\left(R\right)$ that satisfies Equation (24) is called a solution of the strict $k\left[S\right]$-hierarchy in the setting $(R,\left\{{\partial }_i\right\})$. By the same argument as for the $k\left[S\right]$ case, it suffices to prove Equation (24) for $m=1$, since all basis elements $\{{\mathcal{M}}^m\mid m⩾1\}$ of the wider deformation $k\left[\mathcal{M}\right]$ of $k\left[S\right]$ satisfy the same Lax equations. Note that in each setting there is at least one solution of the strict $k\left[S\right]$-hierarchy, namely $\mathcal{M}=S$, the trivial solution of the strict $k\left[S\right]$-hierarchy.

Next, we discuss the Sato–Wilson equations for a ${\mathcal{P}}_{-}$-deformation $\mathcal{M}=PS{P}^{-1}$, with $P=d^{-1}U$, where $d\in {\mathcal{D}}_{\mathbb{N}}{\left(R\right)}^{\ast }$ and $U\in {\mathcal{U}}_{-}$. The analogues for P of Equation (21) are

$${\partial }_i\left(P\right)P^{-1}=-{\pi }_{lt}\left({\mathcal{M}}^i\right).$$

(25)

These are called the Sato–Wilson equations for P. If P satisfies Equation (25), then

$${\partial }_i\left(\mathcal{M}\right)=[{\partial }_i\left(P\right)P^{-1},\mathcal{M}]=[-{\pi }_{lt}\left({\mathcal{M}}^i\right),\mathcal{M}]=[{\mathcal{C}}_i\left(S\right),\mathcal{M}]$$

yields the Lax Equation (24) for $\mathcal{M}$. Hence, Equations (25) are sufficient for $\mathcal{M}$ to be a solution of the strict $k\left[S\right]$-hierarchy. They have a characterization in terms of the components d and U

Proposition 1.

Take a ${\mathcal{P}}_{-}$-deformation $\mathcal{M}=PS{P}^{-1}$, with $P=d^{-1}U$, and let $\mathcal{L}=US{U}^{-1}$. Then, the Sato–Wilson equations for P are equivalent with $\mathcal{L}$ being a solution of the $k\left[S\right]$-hierarchy and d is a solution of the system

$${\partial }_i\left(d\right)=d_0\left({\mathcal{L}}^i\right)d,\mathit{for}\mathit{all}i⩾1,$$

(26)

which is a compatible systems of equations if $\mathcal{L}$ is a solution of the $k\left[S\right]$-hierarchy, see (22).

Proof. 

We express ${\partial }_i\left(P\right)P^{-1}$ in the components d and U

$$\begin{array}{cc}\hfill {\partial }_i\left(P\right)P^{-1}& ={\partial }_i\left(d^{-1}\right)U{U}^{-1}d+d^{-1}{\partial }_i\left(U\right)U^{-1}d\hfill \\ \hfill & =-{\partial }_i\left(d\right)d^{-1}+d^{-1}{\partial }_i\left(U\right)U^{-1}d.\hfill \end{array}$$

Since ${\partial }_i\left(U\right)U^{-1}$ has only strict negative diagonals, the zero-th diagonal of ${\partial }_i\left(P\right)P^{-1}$ is $-{\partial }_i\left(d\right)d^{-1}$ and the sum of all strict negative diagonals of ${\partial }_i\left(P\right)P^{-1}$ is $d^{-1}{\partial }_i\left(U\right)U^{-1}d$. Similarly, we decompose

$$\begin{array}{cc}\hfill -{\pi }_{lt}\left({\mathcal{M}}^i\right)=-{\pi }_{lt}\left(d^{-1}{\mathcal{L}}^{i}d\right)& =-d_0\left(d^{-1}{\mathcal{L}}^{i}d\right)-{\pi }_{slt}\left(d^{-1}{\mathcal{L}}^{i}d\right)\hfill \\ \hfill & =-d_0\left({\mathcal{L}}^i\right)-d^{-1}{\pi }_{slt}\left({\mathcal{L}}^i\right)d.\hfill \end{array}$$

Hence, the zero-th diagonal parts of ${\partial }_i\left(P\right)P^{-1}$ and $-{\pi }_{lt}\left({\mathcal{M}}^i\right)$ are equal if and only if d satisfies the system (26) and the equality of the sum of all their strict negative diagonals is equivalent with the Sato–Wilson equations for $\mathcal{L}$. □

Remark 2.

Note that the choice of the dressing operator P of $\mathcal{M}$ is crucial in (25). Assume that for $P=d^{-1}U$ Equation (25) holds. Then, $\mathcal{M}$ is anyway a solution of the strict $k\left[S\right]$-hierarchy. We have seen that all $P_r=d^{-1}{i}_1\left(r^{-1}\right)U,r\in R^{\ast },$ are also dressing operators of $\mathcal{M}$. The right-hand side of Equation (25) is the same for all dressing operators $P_r$. However, the left-hand side is equal to

$$-{\partial }_i\left(d\right)d^{-1}-i_1\left({\partial }_i\left(r\right)\right)i_1\left(r^{-1}\right)+d^{-1}{\partial }_i\left(U\right)U^{-1}d$$

and that is only equal to the right-hand side if ${\partial }_i\left(r\right)=0$ for all $i⩾1$. In other words, r should be a constant for all the $\left\{{\partial }_i\right\}$. Hence, if you would pick out a dressing operator $P_r$ with ${\partial }_i\left(r\right)\ne 0$ for some i, then the Sato–Wilson equations would not hold for that $P_r$. Nevertheless, the Lax equations of the strict KP hierarchy hold for $\mathcal{M}$.

4. Wave Matrices

Let $(R,\left\{{\partial }_i\right\})$ be a setting where one looks for solutions to both hierarchies. The homogeneous spaces of solutions of both hierarchies are constructed by producing special vectors, called wave matrices, in a suitable $L{T}_{\mathbb{N}}\left(R\right)$-module that we briefly recall. We start with the upper triangular matrix

$${\psi }_0={\psi }_0\left(t\right)=exp(\sum _{i=1}^{\infty }{t}_i{S}^i)=\left(\begin{array}{cccc}1& p_1\left(t\right)& p_2\left(t\right)& \cdots \\ 0& 1& p_1\left(t\right)& \cdots \\ 0& 0& 1& \ddots \\ ⋮& ⋮& \ddots & \ddots \end{array}\right)=\sum _{i=0}^{\infty }{p}_i\left(t\right)S^i.$$

(27)

Here, t is the shorthand notation for $\{t_i\mid i⩾1\}$, $p_0\left(t\right)=1$ and each $p_j\left(t\right),j>0,$ is a homogeneous polynomial of degree j in the $\{t_i\mid i⩽j\}$, where each $t_i$ has degree i. Note that ${\psi }_0$ commutes with S and satisfies for all $i⩾1$, ${\displaystyle \frac{\partial }{\partial t_i}}\left({\psi }_0\right)=S^i{\psi }_0$. Recall that each ${\partial }_i$ was the algebraic substitute on R for the partial derivative with respect to the flow parameter of $S^i$. Therefore, we write ${\partial }_i\left({\psi }_0\right)={\displaystyle \frac{\partial }{\partial t_i}}\left({\psi }_0\right)$. The $L{T}_{\mathbb{N}}\left(R\right)$-module that we need consists of formal products of a perturbation factor from $L{T}_{\mathbb{N}}\left(R\right)$ and ${\psi }_0$. The products will be formal, for making sense out of the product of a matrix from $L{T}_{\mathbb{N}}\left(R\right)$ and the matrix ${\psi }_0$ as a matrix requires convergence conditions and we want to give an algebraic description of the module. Consider therefore the space $\mathcal{O}\left(S\right)$ consisting of the formal products

$$\left\{\left\{m\left(S\right)\right\}{\psi }_0=\left\{\sum _{i⩾0}{m}_i{S}^i+\sum _{i<0}{\left(S^T\right)}^{-i}{m}_i\right\}{\psi }_0,m_i\in {\mathcal{D}}_{\mathbb{N}}\left(R\right)\right\}.$$

Addition (resp. scalar multiplication) is defined on $\mathcal{O}\left(S\right)$ by adding up the perturbation factors of two elements (resp. by applying the scalar multiplication on the perturbation factor). Something similar is done with the $L{T}_{\mathbb{N}}\left(R\right)$-module structure: for each $h\left(S\right)\in$$L{T}_{\mathbb{N}}\left(R\right)$, define

$$h\left(S\right).\left\{m\left(S\right)\right\}{\psi }_0:=\left\{h\left(S\right)m\left(S\right)\right\}{\psi }_0.$$

Clearly, this makes $\mathcal{O}\left(S\right)$ into a free $L{T}_{\mathbb{N}}\left(R\right)$-module with generator ${\psi }_0$. Besides the $L{T}_{\mathbb{N}}\left(R\right)$-action, each ${\partial }_i$ also acts on $\mathcal{O}\left(S\right)$ by the formula

$$\begin{array}{c}\hfill {\partial }_i\left(\left\{m\left(S\right)\right\}{\psi }_0\right):=\left\{\sum _{k=0}^N{\partial }_i\left(m_k\right)S^k+\sum _{k<0}{\left(S^T\right)}^{-k}{\partial }_i\left(m_k\right)+m\left(S\right)S^i\right\}{\psi }_0.\end{array}$$

Here, we impose a Leibniz rule on the formal product. Finally there is a right-hand action of S on $\mathcal{O}\left(S\right)$. Since S and ${\psi }_0$ commute, we can define it by

$$\left\{m\left(S\right)\right\}{\psi }_{0}S:=\left\{m\left(S\right)S\right\}{\psi }_0$$

Analogous to the terminology in the function case, see, e.g., Ref. [28], we call the elements of $\mathcal{O}\left(S\right)$ oscillating $\mathbb{N}\times \mathbb{N}$-matrices. Note that any $\psi =\widehat{\psi }{\psi }_0=h\left(S\right){\psi }_0$ with $h\left(S\right)$ invertible is a generator of the free $L{T}_{\mathbb{N}}\left(R\right)$-module $\mathcal{O}\left(S\right)$. Examples are the choices $h\left(S\right)\in {\mathcal{P}}_{-}$ resp. $h\left(S\right)\in {\mathcal{U}}_{-}$ in which case we call $\psi$ an oscillating $\mathbb{N}\times \mathbb{N}$-matrix of type ${\mathcal{P}}_{-}$ resp. ${\mathcal{U}}_{-}$. With the three actions just defined, we can introduce inside $L{T}_{\mathbb{N}}\left(R\right)$ two systems of equations leading to solutions of the respective hierarchies.

For the $k\left[S\right]$-hierarchy, this system is as follows: consider a ${\mathcal{U}}_{-}$-deformation$\mathcal{L}$ of S in $L{T}_{\mathbb{N}}\left(R\right)$ with the set of projections $\{{\mathcal{B}}_i\left(S\right):={\pi }_{ut}\left({\mathcal{L}}^i\right)\}$. The goal is now to find an oscillating $\mathbb{N}\times \mathbb{N}$-matrix $\psi =\left\{h\left(S\right)\right\}{\psi }_0$ of type ${\mathcal{U}}_{-}$ such that in $\mathcal{O}\left(S\right)$ the following set of equations holds

$$\mathcal{L}\psi =\psi S\mathrm{and}{\partial }_i\left(\psi \right)={\mathcal{B}}_i\left(S\right)\psi ,\mathrm{for}\mathrm{all}i⩾1.$$

(28)

Since $\mathcal{O}\left(S\right)$ is a free $L{T}_{\mathbb{N}}\left(R\right)$-module with generator $\psi$, the first equation $\mathcal{L}\psi =\psi S$ implies $\mathcal{L}h\left(S\right)=h\left(S\right)S$ and thus $\mathcal{L}=h\left(S\right)Sh{\left(S\right)}^{-1}$. By Lemma 2, this determines $h\left(S\right)$ uniquely. The same argument allows one to translate each ${\partial }_i\left(\psi \right)={\mathcal{B}}_i\left(S\right)\psi$ into an identity in $L{T}_{\mathbb{N}}\left(R\right)$:

$${\partial }_i\left(\psi \right)=\{{\partial }_i\left(h\left(S\right)\right)+h\left(S\right)S^i\}{\psi }_0=\{{\partial }_i\left(h\left(S\right)\right)h{\left(S\right)}^{-1}+{\mathcal{L}}^i\}\psi ={\mathcal{B}}_i\left(S\right)\psi$$

Thus, we obtain that $h\left(S\right)$ satisfies the Sato–Wilson Equation (21) and $\mathcal{L}$ is a solution of the $k\left[S\right]$-hierarchy. The system (28) is called the linearization of the $k\left[S\right]$-hierarchy and $\psi$ a wave matrix of the $k\left[S\right]$-hierarchy. Note that ${\psi }_0$ is the wave matrix corresponding to the trivial solution of the $k\left[S\right]$-hierarchy, $\mathcal{L}=S$.

In the case of the strict $k\left[S\right]$-hierarchy, we start with a ${\mathcal{P}}_{-}$-deformation$\mathcal{M}$ of S together with the projections $\{{\mathcal{C}}_i\left(S\right):={\pi }_{sut}\left({\mathcal{M}}^i\right)\}$. Now, we look for an oscillating $\mathbb{N}\times \mathbb{N}$-matrix $\phi =\left\{k\left(S\right)\right\}{\psi }_0$ of type ${\mathcal{P}}_{-}$ that satisfies in $\mathcal{O}\left(S\right)$ the following set of equations:

$$\mathcal{M}\phi =\phi S\mathrm{and}{\partial }_i\left(\phi \right)={\mathcal{C}}_i\left(S\right)\phi ,\mathrm{for}\mathrm{all}i⩾1.$$

(29)

Also, $\phi$ is a generator of $\mathcal{O}\left(S\right)$ and again we can translate Equation (29) into identities in $L{T}_{\mathbb{N}}\left(R\right)$. Thus, the first equation $\mathcal{M}\phi =\phi S$ becomes $\mathcal{M}=k\left(S\right)Sk{\left(S\right)}^{-1}$ and the second set of equations in (29) yields the Sato–Wilson Equation (25) of the strict $k\left[S\right]$-hierarchy. In particular, $\mathcal{M}$ is a solution of that hierarchy. The system (29) is called the linearization of the strict $k\left[S\right]$-hierarchy and a $\phi$ satisfying this system a wave matrix of the strict $k\left[S\right]$-hierarchy.

For both hierarchies, we use in the sequel a milder property that oscillating $\mathbb{N}\times \mathbb{N}$-matrices of a certain type have to satisfy in order to become a wave matrix of that hierarchy. For a proof, see [41].

Proposition 2.

Let $\psi =\left\{h\left(S\right)\right\}{\psi }_0$ be an oscillating $\mathbb{N}\times \mathbb{N}$-matrix of type ${\mathcal{U}}_{-}$ in $\mathcal{O}\left(S\right)$ and $\mathcal{L}=h\left(S\right)Sh{\left(S\right)}^{-1}$ the corresponding potential solution of the $k\left[S\right]$-hierarchy. Similarly, let $\phi =\left\{k\left(S\right)\right\}{\psi }_0$ be an oscillating $\mathbb{N}\times \mathbb{N}$-matrix of type ${\mathcal{P}}_{-}$ in $\mathcal{O}\left(S\right)$ with potential solution $\mathcal{M}=k\left(S\right)Sk{\left(S\right)}^{-1}$ of the strict version.

(a) 

Assume there exists for each $i⩾1$ an element $M_i\in {\pi }_{ut}\left(L{T}_{\mathbb{N}}\left(R\right)\right)$ such that

$${\partial }_i\left(\psi \right)=M_i\psi .$$

Then, each $M_i={\pi }_{ut}\left({\mathcal{L}}^i\right)$ and ψ is a wave matrix for the $k\left[S\right]$-hierarchy.

(b) 

Suppose there exists for each $i⩾1$ an element $N_i\in {\pi }_{sut}\left(L{T}_{\mathbb{N}}\left(R\right)\right)$ such that

$${\partial }_i\left(\phi \right)=N_i\phi .$$

Then, each $N_i={\pi }_{sut}\left({\mathcal{M}}^i\right)$ and φ is a wave matrix for the strict $k\left[S\right]$-hierarchy.

Since you do not meet formal products of lower triangular $\mathbb{N}\times \mathbb{N}$-matrices and upper triangular $\mathbb{N}\times \mathbb{N}$-matrices in real life, the only way to construct wave matrices of both hierarchies is to give an analytic framework, where you can produce well-defined products of such matrices. This is done in the next section.

5. LU Factorizations and Solutions of Both Hierarchies

All the relevant $\mathbb{N}\times \mathbb{N}$-matrices that will be produced in the sequel correspond to bounded operators acting on a separable real or complex Banach space with a countable Schauder basis. Since $k=\mathbb{R}$ or $\mathbb{C}$, we have on each k the standard norm $|·|:k\to {\mathbb{R}}_{⩾0}.$ The Banach spaces B we will work with are different spaces of $\mathbb{N}\times 1$ matrices. In all cases the Schauder basis consists of the $\{\overrightarrow{e}\left(n\right)\mid n\in \mathbb{N}\}$ introduced in Section 2. The choice we make for B is either one of the spaces $B\left(p\right),1⩽p<\infty ,$ of $\mathbb{N}\times 1$ matrices, defined by

$$B\left(p\right)=\{b\mid b^T=(b_0,b_1,b_2,\cdots )\mid b_n\in k,\sum _{n\in \mathbb{N}}|b_n{|}^p<\infty \},$$

with the norm

$${\left|\right|b\left|\right|}_p=(\sum _{n\in \mathbb{N}}{\left|b_n{|}^p\right)}^{{\displaystyle \frac{1}{p}}}$$

or the space $B\left(0\right)$ defined by

$$B\left(0\right)=\{b\mid b^T=\sum _{n\in \mathbb{N}}{b}_n\overrightarrow{e}{\left(n\right)}^T=(b_0,b_1,b_2,\cdots )\mid b_n\in k,\underset{n\to \infty }{lim}{b}_n=0\},$$

with the supremum norm $\left|\right|·{\left|\right|}_0$. Any $A\in \mathcal{L}\left(B\right)$, the space of all bounded k-linear maps from B to itself, corresponds to left multiplication $M_{\left[A\right]}$ with an $\mathbb{N}\times \mathbb{N}$-matrix $\left[A\right]=\left(a_{ij}\right)$ with respect to $\left\{\overrightarrow{e}\left(n\right)\right\}$, i.e.,

$$A\left(\overrightarrow{e}\left(j\right)\right)=M_{\left[A\right]}\left(\overrightarrow{e}\left(j\right)\right)=\left[A\right]\overrightarrow{e}\left(j\right)=\sum _{i\in \mathbb{N}}{a}_{ij}\overrightarrow{e}\left(i\right).$$

The invertible transformations in $\mathcal{L}\left(B\right)$ are denoted by $GL\left(B\right)$. For example, the operator $M_S$ as defined in Section 2 by Formula (12) maps each of these Banach spaces $B\left(q\right),q\in \left\{0\right\}\cup [1,\infty ),$ to itself and defines an operator in $\mathcal{L}\left(B\right)$ with operator norm equal to 1.

All the Banach spaces $B\left(q\right)$ have two properties in common that we need later on: first of all, they have the approximation property, i.e., the bounded finite dimensional operators in $\mathcal{L}\left(B\right)$ are dense with respect to the operator norm in the compact operators $\mathcal{K}\left(B\right)$ in $\mathcal{L}\left(B\right)$, see, e.g., Ref. [43]. Secondly, each vector in $B\left(q\right)$ is the limit of a Cauchy sequence of vectors with only a finite number of nonzero coordinates with respect to $\left\{\overrightarrow{e}\left(n\right)\right\}$.

Now, we come to the description of the role of LU factorization in $GL\left(B\right)$ for the construction of solutions of both hierarchies. Two decompositions of $\mathcal{L}\left(B\right)$ play a role in this process. The first splits $b\in \mathcal{L}\left(B\right)$ as $b=u_{-}\left(b\right)+p_{+}\left(b\right)$, with the corresponding matrices

$$\left[u_{-}\left(b\right)\right]=\left(\begin{array}{cccc}0& 0& 0& \cdots \\ b_{10}& 0& 0& \cdots \\ b_{20}& b_{21}& 0& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)\mathrm{and}\left[p_{+}\left(b\right)\right]=\left(\begin{array}{cccc}{b}_{00}& b_{01}& b_{02}& \cdots \\ 0& b_{11}& b_{12}& \cdots \\ 0& 0& b_{22}& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right).$$

It gives rise to the decomposition of the Lie algebra $\mathcal{L}\left(B\right)$ as ${\mathcal{U}}_{-}\left(B\right)\oplus {\mathcal{P}}_{+}\left(B\right)$, where

$$\begin{array}{cc}\hfill & {\mathcal{U}}_{-}\left(B\right)=\{b\in \mathcal{L}\left(B\right)\mid b=u_{-}\left(b\right)\}\mathrm{and}{\mathcal{P}}_{+}\left(B\right)=\{b\in \mathcal{L}\left(B\right)\mid b=p_{+}\left(b\right)\}.\hfill \end{array}$$

(30)

The Lie groups corresponding to ${\mathcal{U}}_{-}\left(B\right)$ and ${\mathcal{P}}_{+}\left(B\right)$ are, respectively

$$\begin{array}{cc}\hfill & {\mathbf{U}}_{-}\left(B\right)=\{g\in GL\left(B\right)\mid g-Id\in {\mathcal{U}}_{-}\left(B\right)\}\mathrm{and}\hfill \\ \hfill & {\mathbf{P}}_{+}\left(B\right)=\{g\in GL\left(B\right)\mid g=p_{+}\left(g\right)\}.\hfill \end{array}$$

(31)

Since the exponential map is a local diffeomorphism from an open neighbourhood of 0 in ${\mathcal{U}}_{-}\left(B\right)$ resp. ${\mathcal{P}}_{+}\left(B\right)$ to an open neighbourhood of the identity element in ${\mathbf{U}}_{-}\left(B\right)$ resp. ${\mathbf{P}}_{+}\left(B\right)$, it follows that $\Omega \left(B\right)={\mathbf{U}}_{-}\left(B\right){\mathbf{P}}_{+}\left(B\right)$ is a neighbourhood of the identity in $GL\left(B\right)$. As any point $\omega$ of $\Omega \left(B\right)$ can be reached with left multiplication with elements of ${\mathbf{U}}_{-}\left(B\right)$ and right multiplication with elements of ${\mathbf{P}}_{+}\left(B\right)$, the set $\Omega \left(B\right)$ is open in $GL\left(B\right)$. It is called the big cell in $GL\left(B\right)$ with respect to the groups ${\mathbf{U}}_{-}\left(B\right)$ and ${\mathbf{P}}_{+}\left(B\right)$. Since the groups ${\mathbf{U}}_{-}\left(B\right)$ and ${\mathbf{P}}_{+}\left(B\right)$ intersect only in the identity, the splitting of an $\omega$ from $\Omega \left(B\right)$ in the product of a unipotent lower triangular matrix and an upper triangular matrix is unique. We use a slightly twisted version of this LU factorization of $\Omega \left(B\right)$: each $\omega \in \Omega \left(B\right)$ splits uniquely as

$$\omega ={\mathbf{u}}_{-}{\left(\omega \right)}^{-1}{\mathbf{p}}_{+}\left(\omega \right),\mathrm{with}{\mathbf{u}}_{-}\left(\omega \right)\in {\mathbf{U}}_{-}\left(B\right)\mathrm{and}{\mathbf{p}}_{+}\left(\omega \right)\in {\mathbf{P}}_{+}\left(B\right).$$

(32)

The component ${\mathbf{u}}_{-}\left(\omega \right)$ we call the unipotent component of the LU factorization (32).

The second decomposition is a variation of the foregoing and consists of splitting a $b\in \mathcal{L}\left(B\right)$ as $b=p_{-}\left(b\right)+u_{+}\left(b\right)$, where the matrices of both components are given by

$$\left[p_{-}\left(b\right)\right]=\left(\begin{array}{cccc}{b}_{00}& 0& 0& \cdots \\ b_{10}& b_{11}& 0& \cdots \\ b_{20}& b_{21}& b_{22}& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right)\mathrm{and}\left[u_{+}\left(b\right)\right]=\left(\begin{array}{cccc}0& b_{01}& b_{02}& \cdots \\ 0& 0& b_{12}& \cdots \\ 0& 0& 0& \cdots \\ ⋮& ⋮& ⋮& \ddots \end{array}\right).$$

This leads to the decomposition $\mathcal{L}\left(B\right)={\mathcal{P}}_{-}\left(B\right)\oplus {\mathcal{U}}_{+}\left(B\right)$ of $\mathcal{L}\left(B\right)$, where

$$\begin{array}{cc}\hfill & {\mathcal{P}}_{-}\left(B\right)=\{b\in \mathcal{L}\left(B\right)\mid b=p_{-}\left(b\right)\}\mathrm{and}{\mathcal{U}}_{+}\left(B\right)=\{b\in \mathcal{L}\left(B\right)\mid b=u_{+}\left(b\right)\}.\hfill \end{array}$$

(33)

The Lie groups corresponding to ${\mathcal{P}}_{-}\left(B\right)$ and ${\mathcal{U}}_{+}\left(B\right)$ are, respectively

$$\begin{array}{cc}\hfill & {\mathbf{P}}_{-}\left(B\right)=\{g\in GL\left(B\right)\mid g=p_{-}\left(g\right)\}\mathrm{and}\hfill \\ \hfill & {\mathbf{U}}_{+}\left(B\right)=\{g\in GL\left(B\right)\mid g-Id\in {\mathcal{U}}_{+}\left(B\right)\}.\hfill \end{array}$$

(34)

Let $\mathbf{D}\left(B\right)$ denote the diagonal matrices in ${\mathbf{P}}_{+}\left(B\right)$. Then, we have ${\mathbf{P}}_{+}\left(B\right)=\mathbf{D}\left(B\right){\mathbf{U}}_{+}\left(B\right)$ and the big cell ${\mathbf{P}}_{-}\left(B\right){\mathbf{U}}_{+}\left(B\right)$ with respect to ${\mathbf{P}}_{-}\left(B\right)$ and ${\mathbf{U}}_{+}\left(B\right)$ is also equal to $\Omega \left(B\right)$. This yields an alternative LU factorization for an element in $\Omega \left(B\right)$ in the product of a lower triangular matrix and a unipotent upper triangular matrix. We will again use a twisted version of this factorization: each $\omega$ in $\Omega \left(B\right)$ splits uniquely as follows:

$$\omega ={\mathbf{p}}_{-}{\left(\omega \right)}^{-1}{\mathbf{u}}_{+}\left(\omega \right),\mathrm{with}{\mathbf{p}}_{-}\left(\omega \right)\in {\mathbf{P}}_{-}\left(B\right)\mathrm{and}{\mathbf{u}}_{+}\left(\omega \right)\in {\mathbf{P}}_{+}\left(B\right).$$

(35)

The component ${\mathbf{p}}_{-}\left(\omega \right)$ we call the parabolic component of the LU factorization (35).

Basically, the group of commuting flows that is relevant for both hierarchies is described in the generator ${\psi }_0$ of the module $\mathcal{O}\left(S\right)$ of oscillating matrices. What needs to be done still is to make a proper choice for the $\left\{t_i\right\}$ such that for those parameters ${\psi }_0\left(t\right)$ is the matrix of an element in $GL\left(B\right)$. We have seen that the operator norm $\left|\right|M_S\left|\right|$ is equal to 1. Hence, the following choice defines a subgroup $\Gamma$ in $GL\left(B\right)$:

$$\begin{array}{c}\hfill \Gamma =\left\{\gamma =exp\left(\sum _{i=1}^{\infty }{t}_i{M}_S^i\right)\mid \begin{array}{c}\sum _{i=1}^{\infty }\left|t_i\right|{(1+\epsilon )}^i<\infty \\ \mathrm{for}\mathrm{some}\epsilon >0\end{array}\right\}.\end{array}$$

(36)

For each $\gamma \in \Gamma ,$ there holds $\left[\gamma \right]={\sum }_{i=0}^{\infty }{p}_i\left(t\right)S^i$, with the polynomials $\left\{p_i\left(t\right)\right\}$ as in (27), and $\Gamma$ is thus a subgroup of ${\mathbf{U}}_{+}\left(B\right)$.

Now, we construct for each g in the open set $\Gamma \Omega \left(B\right)$ a solution of the $k\left[S\right]$-hierarchy and the strict $k\left[S\right]$-hierarchy. For each $g\in GL\left(B\right)$, consider the open subset $\Gamma \left(g\right)$ of $\Gamma$ defined by

$$\Gamma \left(g\right)=\{\gamma \in \Gamma \mid \gamma g\in \Omega (B\left)\right\}.$$

The subset $\Gamma \left(g\right)$ is nonempty if and only if $g\in \Gamma \Omega \left(B\right)$, which we assume from now on. The appropriate setting in both cases is the algebra

$$R\left(g\right)=C^{\infty }(\Gamma \left(g\right),k),$$

with the derivations ${\partial }_i={\displaystyle \frac{\partial }{\partial t_i}},i⩾1.$ We start with the construction of the solutions of the $k\left[S\right]$-hierarchy. Then, we have by definition for all $\gamma \in \Gamma \left(g\right)$ that

$$\gamma g={\mathbf{u}}_{-}{(g,\gamma )}^{-1}{\mathbf{p}}_{+}(g,\gamma )$$

(37)

and thus on the matrix level

$$\left[\gamma \right]\left[g\right]={\left[{\mathbf{u}}_{-}(g,\gamma )\right]}^{-1}\left[{\mathbf{p}}_{+}(g,\gamma )\right]$$

Note that all matrix coefficients of $\left[{\mathbf{u}}_{-}(g,\gamma )\right]$ and $\left[{\mathbf{p}}_{+}(g,\gamma )\right]$ belong to $R\left(g\right)$, since the map $({\mathbf{u}}_{-},{\mathbf{p}}_{+})\to {\mathbf{u}}_{-}^{-1}{\mathbf{p}}_{+}$ is a diffeomorphism between ${\mathbf{U}}_{-}\left(B\right)\times {\mathbf{P}}_{+}\left(B\right)$ and $\Omega \left(B\right)$. Equation (37) leads to the following identity

$$\psi \left(g\right):=\left[{\mathbf{u}}_{-}(g,\gamma )\right]\left[\gamma \right]=\left[{\mathbf{p}}_{+}(g,\gamma )\right]{\left[g\right]}^{-1}$$

(38)

Clearly, $\psi \left(g\right)$ is an oscillating matrix in $\mathcal{O}\left(S\right)$ for which the products between the different factors are real. To show that $\psi \left(g\right)$ is a wave matrix for the $k\left[S\right]$-hierarchy, it suffices to prove the property in Proposition 2. Thus, we compute for all $i⩾1$, the matrix ${\partial }_i\left(\psi \left(g\right)\right)\psi {\left(g\right)}^{-1}$ using both the left- and the right-hand sides of expression (38). We start with the right-hand side. Since for all $i⩾1$, ${\partial }_i\left(\left[g\right]\right)=0$, we obtain

$$\begin{array}{c}{\partial }_i\left(\psi \left(g\right)\right)\psi {\left(g\right)}^{-1}={\partial }_i\left(\left[{\mathbf{p}}_{+}(g,\gamma )\right]\right){\left[{\mathbf{p}}_{+}(g,\gamma )\right]}^{-1}\end{array}$$

Now, the matrix ${\partial }_i\left(\left[{\mathbf{p}}_{+}(g,\gamma )\right]\right){\left[{\mathbf{p}}_{+}(g,\gamma )\right]}^{-1}$ is of the form ${\sum }_{r⩾0}{d}_r{S}^r$ with all $d_r\in {\mathcal{D}}_{\mathbb{N}}\left(R\left(g\right)\right)$. Next, we use the left-hand side of (38) to compute ${\partial }_i\left(\psi \left(g\right)\right)\psi {\left(g\right)}^{-1}$. This yields

$$\begin{array}{cc}\hfill {\partial }_i\left(\psi \left(g\right)\right)\psi {\left(g\right)}^{-1}& ={\partial }_i\left(\left[{\mathbf{u}}_{-}(g,\gamma )\right]\right){\left[{\mathbf{u}}_{-}(g,\gamma )\right]}^{-1}+\left[{\mathbf{u}}_{-}(g,\gamma )\right]{\partial }_i\left(\left[\gamma \right]\right){\left[\gamma \right]}^{-1}{\left[u_{-}(g,\gamma )\right]}^{-1}\hfill \\ \hfill & ={\partial }_i\left(\left[{\mathbf{u}}_{-}(g,\gamma )\right]\right){\left[u_{-}(g,\gamma )\right]}^{-1}+\left[{\mathbf{u}}_{-}(g,\gamma )\right]S^i{\left[{\mathbf{u}}_{-}(g,\gamma )\right]}^{-1}\hfill \end{array}$$

In this formula, expression ${\partial }_i\left(\left[{\mathbf{u}}_{-}(g,\gamma )\right]\right){\left[{\mathbf{u}}_{-}(g,\gamma )\right]}^{-1}$ possesses only negative diagonals and $\left[{\mathbf{u}}_{-}(g,\gamma )\right]S^i{\left[{\mathbf{u}}_{-}(g,\gamma )\right]}^{-1}$ has the form

$$\sum _{r=0}^i{v}_r{S}^r+\sum _{r<0}{\left(S^T\right)}^{-r}{v}_r,$$

with all $v_r\in {\mathcal{D}}_{\mathbb{N}}\left(R\right)$ and $v_i=Id$. Combining this with the expression found for the right-hand side gives for all $i⩾1$

$${\partial }_i\left(\psi \left(g\right)\right)=(\sum _{r=0}^i{v}_r{S}^r)\psi \left(g\right)={\mathcal{B}}_{i,\psi \left(g\right)}\psi \left(g\right).$$

Thus, $\psi \left(g\right)$ satisfies the conditions in part (a) of Proposition 2 and hence it is a wave matrix of the $k\left[S\right]$-hierarchy. In other words, $\psi \left(g\right)$ is a solution of the linearization of the $k\left[S\right]$-hierarchy. The corresponding solution $\mathcal{L}\left(g\right)$ of the $k\left[S\right]$-hierarchy is

$$\mathcal{L}\left(g\right)=\left[u_{-}(g,\gamma )\right]S{\left[u_{-}(g,\gamma )\right]}^{-1}$$

(39)

and ${\mathcal{B}}_{i,\psi \left(g\right)}={\pi }_{ut}\left(\mathcal{L}{\left(g\right)}^i\right)$. Note that, since the factor $p_{+}{(g,\gamma )}^{-1}$ plays no role in the construction of $\mathcal{L}\left(g\right)$, multiplying g from the right with an element of ${\mathbf{P}}_{+}\left(B\right)$ does not affect the solution $\mathcal{L}\left(g\right)$.

Secondly, we present for a $g\in \Gamma \Omega \left(B\right)$ the construction of the solution of the strict $k\left[S\right]$-hierarchy. We proceed similarly, but now we use the LU factorization (35) of $\Omega \left(B\right)$. By definition, we have for all $\gamma \in \Gamma \left(g\right)$ that

$$\gamma g={\mathbf{p}}_{-}{(g,\gamma )}^{-1}{\mathbf{u}}_{+}(g,\gamma )$$

(40)

and thus on the matrix level

$$\left[\gamma \right]\left[g\right]={\left[{\mathbf{p}}_{-}(g,\gamma )\right]}^{-1}\left[{\mathbf{u}}_{+}(g,\gamma )\right]$$

Note that all matrix coefficients of $\left[{\mathbf{p}}_{-}(g,\gamma )\right]$ and $\left[{\mathbf{u}}_{+}(g,\gamma )\right]$ belong to $R\left(g\right)$, since the map $({\mathbf{p}}_{-},{\mathbf{u}}_{+})\to {\mathbf{p}}_{-}^{-1}{\mathbf{u}}_{+}$ is a diffeomorphism between ${\mathbf{P}}_{-}\left(B\right)\times {\mathbf{U}}_{+}\left(B\right)$ and $\Omega \left(B\right)$. Equation (40) leads to the following identity

$$\phi \left(g\right):=\left[{\mathbf{p}}_{-}(g,\gamma )\right]\left[\gamma \right]=\left[{\mathbf{u}}_{+}(g,\gamma )\right]{\left[g\right]}^{-1}$$

(41)

Clearly, $\phi \left(g\right)$ is an oscillating matrix in $\mathcal{O}\left(S\right)$ for which the product between the different factors is real. The idea is again to show that $\phi \left(g\right)$ is a wave matrix for the strict $k\left[S\right]$-hierarchy and that is done by proving property (b) in Proposition 2. Thus, we compute for all $i⩾1$, the matrix ${\partial }_i\left(\phi \left(g\right)\right)\phi {\left(g\right)}^{-1}$ using both the left- and the right-hand sides of expression (41). We start with the right-hand side. Again all the ${\partial }_i\left(\left[g\right]\right)$ are zero; hence,

$$\begin{array}{cc}\hfill {\partial }_i\left(\phi \left(g\right)\right)\phi {\left(g\right)}^{-1}& ={\partial }_i\left(\left[{\mathbf{u}}_{+}(g,\gamma )\right]\right){\left[{\mathbf{u}}_{+}(g,\gamma )\right]}^{-1}\hfill \end{array}$$

The matrix ${\partial }_i\left(\left[u_{+}(g,\gamma )\right]\right){\left[u_{+}(g,\gamma )\right]}^{-1}$ only has strict positive diagonals. Thus, the expression ${\partial }_i\left(\phi \left(g\right)\right)\phi {\left(g\right)}^{-1}$ is equal to a matrix of the form ${\sum }_{r⩾1}{u}_r{S}^r$ with all $u_r\in {\mathcal{D}}_{\mathbb{N}}\left(R\left(g\right)\right)$. Next, we use the left-hand side of (41) to compute ${\partial }_i\left(\phi \left(g\right)\right)\phi {\left(g\right)}^{-1}$ once more. This yields

$$\begin{array}{cc}\hfill {\partial }_i\left(\phi \left(g\right)\right)\phi {\left(g\right)}^{-1}& ={\partial }_i\left(\left[{\mathbf{p}}_{-}(g,\gamma )\right]\right){\left[{\mathbf{p}}_{-}(g,\gamma )\right]}^{-1}+\left[{\mathbf{p}}_{-}(g,\gamma )\right]{\partial }_i\left(\left[\gamma \right]\right){\left[\gamma \right]}^{-1}{\left[{\mathbf{p}}_{-}(g,\gamma )\right]}^{-1}\hfill \\ \hfill & ={\partial }_i\left(\left[{\mathbf{p}}_{-}(g,\gamma )\right]\right){\left[{\mathbf{p}}_{-}(g,\gamma )\right]}^{-1}+\left[{\mathbf{p}}_{-}(g,\gamma )\right]S^i{\left[{\mathbf{p}}_{-}(g,\gamma )\right]}^{-1}\hfill \end{array}$$

In this formula, the expression ${\partial }_i\left(\left[{\mathbf{p}}_{-}(g,\gamma )\right]\right){\left[{\mathbf{p}}_{-}(g,\gamma )\right]}^{-1}$ does not possess any strict positive diagonals and the matrix $\left[{\mathbf{p}}_{-}(g,\gamma )\right]S^i{\left[{\mathbf{p}}_{-}(g,\gamma )\right]}^{-1}$ has the form

$$\sum _{r=0}^i{v}_r{S}^r+\sum _{r<0}{\left(S^T\right)}^{-r}{v}_r,$$

with all $v_r\in {\mathcal{D}}_{\mathbb{N}}\left(R\right)$ and $v_i\in {\mathcal{D}}_{\mathbb{N}}{\left(R\right)}^{\ast }$. Combining this with the first expression found yields for all $i⩾1$

$${\partial }_i\left(\phi \left(g\right)\right)=(\sum _{r=1}^i{v}_r{S}^r)\phi \left(g\right)={\mathcal{C}}_{i,\phi \left(g\right)}\phi \left(g\right).$$

Thus, $\phi \left(g\right)$ satisfies the conditions in part (b) of Proposition 2 and hence is a wave matrix of the strict $k\left[S\right]$-hierarchy. The corresponding solution $\mathcal{M}\left(g\right)$ of the strict $k\left[S\right]$-hierarchy is

$$\mathcal{M}\left(g\right)=\left[{\mathbf{p}}_{-}(g,\gamma )\right]S{\left[{\mathbf{p}}_{-}(g,\gamma )\right]}^{-1}$$

(42)

and ${\mathcal{C}}_{i,\phi \left(g\right)}={\pi }_{sut}\left(\mathcal{M}{\left(g\right)}^i\right)$. Also, here the factor ${\mathbf{u}}_{+}{(g,\gamma )}^{-1}$ plays no role at the construction of $\mathcal{M}\left(g\right)$. Hence, multiplying g from the right with an element of ${\mathbf{U}}_{+}\left(B\right)$ does not affect the solution $\mathcal{M}\left(g\right)$. For completeness, we resume the foregoing results.

Theorem 1.

Let g be an element in the open set $\Gamma \Omega \left(B\right)$.

(a) 

For any γ in $\Gamma \left(g\right)$, let ${\mathbf{u}}_{-}(g,\gamma )$ be the unipotent component of $\gamma g$ in the LU factorization (32) of $\Omega \left(B\right)$. Let the oscillating matrix $\psi \left(g\right)\in \mathcal{O}\left(S\right)$ be defined by Formula (38). Then, $\psi \left(g\right)$ is the wave matrix of the $k\left[S\right]$-hierarchy with respect to the matrix $\mathcal{L}\left(g\right)$ defined by Formula (39). The solution $\mathcal{L}\left(g\right)$ of the $k\left[S\right]$-hierarchy satisfies for all $g\in \Gamma \Omega \left(B\right)$ and all $p\in {\mathbf{P}}_{+}\left(B\right)$ that $\mathcal{L}\left(g\right)=\mathcal{L}\left(gp\right)$.

(b) 

For any γ in $\Gamma \left(g\right)$, let ${\mathbf{p}}_{-}(g,\gamma )$ be the parabolic component of $\gamma g$ in the LU factorization (35) of $\Omega \left(B\right)$. Let the oscillating matrix $\phi \left(g\right)\in \mathcal{O}\left(S\right)$ be defined by Formula (41). Then, $\phi \left(g\right)$ is the wave matrix of the strict $k\left[S\right]$-hierarchy with respect to the matrix ${\mathcal{M}}_g$ defined by Formula (42). The solution $\mathcal{M}\left(g\right)$ of the strict $k\left[S\right]$-hierarchy satisfies for all $g\in \Gamma \Omega \left(B\right)$ and all $u\in {\mathbf{U}}_{+}\left(B\right)$ that $\mathcal{M}\left(g\right)=\mathcal{M}\left(gu\right)$.

In the sequel, we need the decomposition $B=B_{⩽i}\oplus B_{>i}$, where the subspaces $\{B_{⩽i},i\in \mathbb{N}\}$ of B and their complements $B_{>i}$ are defined by

$$B_{⩽i}=\{\sum _{n⩽i}{b}_n\overrightarrow{e}\left(n\right)\in B\}\mathrm{and}{B}_{>i}=\{\sum _{n>i}{b}_n\overrightarrow{e}\left(n\right)\in B\}.$$

Each $b\in \mathcal{L}\left(B\right)$ splits as follows with respect to $B=B_{⩽i}\oplus B_{>i}$:

$$b=\left(\begin{array}{cc}{b}_i\left(11\right)& b_i\left(12\right)\\ b_i\left(21\right)& b_i\left(22\right)\end{array}\right)\mathrm{and}\mathrm{its}\mathrm{matrix}\mathrm{as}\left[b\right]=\left(\begin{array}{cc}\left[b_i\left(11\right)\right]& \left[b_i\left(12\right)\right]\\ [b_i\left(21\right)]& \left[b_i\left(22\right)\right]\end{array}\right).$$

(43)

Next, we have a more detailed look at the matrix coefficients of the dressing matrices constructed in Theorem 1. Consider $g\in \Gamma \Omega \left(B\right)$, with $\left[g\right]=\left(g_{i,j}\right)$, and a $\gamma \in \Gamma$. Then, $\left[\gamma \right]\left[g\right]=\left[g\left(t\right)\right]=\left(g_{i,j}\left(t\right)\right)$, with

$$g_{i,j}\left(t\right)=\sum _{k⩾i}{p}_{k-i}\left(t\right)g_{k,j},\mathrm{for}\mathrm{all}i,j\in \mathbb{N}.$$

Hence, all the matrix coefficients $\left\{g_{i,j}\left(t\right)\right\}$ belong to the analytic functions $\mathcal{A}(\Gamma )$ on $\Gamma$. If $\gamma \in \Gamma \left(g\right)$, then $\gamma g\in \Omega \left(B\right)$ and $\gamma g=p_{-}\left(t\right)u_{+}\left(t\right)$ with $p_{-}\left(t\right)\in {\mathbf{P}}_{-}\left(B\right)$ and $u_{+}\in {\mathbf{U}}_{+}\left(B\right)$. In particular, we have

$$\left[p_{-}\left(t\right)\right]=\left(\begin{array}{ccccc}{p}_{0,0}\left(t\right)& 0& \cdots & 0& \cdots \\ ⋮& \ddots & \ddots & \cdots & \ddots \\ ⋮& \ddots & \ddots & 0& \ddots \\ p_{i,0}\left(t\right)& ⋮& \ddots & p_{i,i}\left(t\right)& \ddots \\ ⋮& \ddots & \ddots & \ddots & \ddots \end{array}\right),\left[u_{+}\left(t\right)\right]=\left(\begin{array}{ccccc}1& u_{0,1}\left(t\right)& \cdots & u_{0,i}\left(t\right)& \cdots \\ 0& \ddots & \ddots & \cdots & \ddots \\ ⋮& \ddots & \ddots & u_{i-1,i}\left(t\right)& \ddots \\ 0& \cdots & 0& 1& \ddots \\ ⋮& \ddots & \ddots & \ddots & \ddots \end{array}\right).$$

Using the decomposition (43) for $p_{-}\left(t\right)$ and $u_{+}\left(t\right)$, we obtain for all $n\in \mathbb{N}$ the finite dimensional LU factorization $\left[g{\left(t\right)}_n(1,1)\right]=\left[p_{-}{\left(t\right)}_n(1,1)\right]\left[u_{+}{\left(t\right)}_n(1,1)\right]$, which enables you to express each $p_{i,j}\left(t\right)$ and $u_{i,j}\left(t\right)$ as the quotient of two polynomial expressions in $\{g_{ij}\left(t\right)\mid i⩽n,j⩽n\}$. This is the case for a number of matrix coefficients that can be seen directly: for example, the first column of $p_{-}\left(t\right)$ has to equal that of $g\left(t\right)$. So, $p_{k,0}\left(t\right)=g_{k,0}\left(t\right),k\in \mathbb{N}.$ The LU factorization of $\left[g{\left(t\right)}_n(1,1)\right]$ shows that

$$Q_n=det\left(\left[g{\left(t\right)}_n(1,1)\right]\right)=det\left(\left[p_{-}{\left(t\right)}_n(1,1)\right]\right)=\prod _{k=0}^n{p}_{kk}\left(t\right)\ne 0$$

In particular, we obtain that for $1⩽k⩽n$, $p_{k,k}\left(t\right)=Q_k{Q}_{k-1}^{-1}$. With these data, one can prove by induction on the size of the matrix that all other matrix coefficients of $\left[p_{-}{\left(t\right)}_n(1,1)\right]$ and $\left[u_{+}{\left(t\right)}_n(1,1)\right]$ are determined uniquely and have the mentioned form. This can be seen as follows: consider the matrix

$$\left[g{\left(t\right)}_{n+1}(1,1)\right]=\left(\begin{array}{ccccc}{g}_{0,0}\left(t\right)& \cdots & \cdots & g_{0,n}\left(t\right)& g_{0,n+1}\left(t\right)\\ ⋮& \ddots & \ddots & \cdots & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ g_{n,0}\left(t\right)& ⋮& \ddots & g_{n,n}\left(t\right)& g_{n,n+1}\left(t\right)\\ g_{n+1,0}\left(t\right)& \ddots & \ddots & g_{n+1,n}\left(t\right)& g_{n+1,n+1}\left(t\right)\end{array}\right)$$

Now, all $Q_k,0⩽k⩽n+1,$ are nonzero and we apply the induction hypothesis to the submatrix $\left[g{\left(t\right)}_n(1,1)\right]$ to obtain the unique coefficients $\left\{p_{i,j}\left(t\right)\right\}$ and $\left\{u_{i,j}\left(t\right)\right\}$ with i and j varying between 0 and n. Now, we know already $p_{n+1,n+1}\left(t\right)=Q_{n+1}{Q}_n^{-1}$ and $p_{n+1,0}\left(t\right)=g_{n+1,0}\left(t\right)$ so that we merely have to find $\{u_{kn+1}\left(t\right)\mid 0⩽k⩽n\}$ and $\{p_{n+1k}\left(t\right)\mid 1⩽k⩽n\}$. For the first set, we have

$$\left(\begin{array}{cccc}{p}_{0,0}\left(t\right)& 0& \cdots & 0\\ ⋮& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & 0& \\ p_{n,0}\left(t\right)& ⋮& \ddots & p_{n,n}\left(t\right)\end{array}\right)\left(\begin{array}{c}{u}_{0,n+1}\left(t\right)\\ ⋮\\ ⋮\\ u_{n,n+1}\left(t\right)\end{array}\right)=\left(\begin{array}{c}{g}_{0,n+1}\left(t\right)\\ ⋮\\ ⋮\\ g_{n,n+1}\left(t\right)\end{array}\right)$$

and since $\left[p_{-}{\left(t\right)}_n(1,1)\right]$ is invertible, this determines the first set completely and the whole matrix $\left[u_{+}{\left(t\right)}_{n+1}(1,1)\right]$. The product of the last row of $\left[p_{-}{\left(t\right)}_{n+1}(1,1)\right]$ with the second column of $\left[u_{+}{\left(t\right)}_{n+1}(1,1)\right]$ yields the equation $p_{n+1,0}\left(t\right)u_{0,1}\left(t\right)+p_{n+1,1}\left(t\right)=g_{n+1,1}\left(t\right)$ and that determines $p_{n+1,1}\left(t\right)$ uniquely. From the product of the last row of $\left[p_{-}{\left(t\right)}_{n+1}(1,1)\right]$ with the third column of $\left[u_{+}{\left(t\right)}_{n+1}(1,1)\right]$ one obtains the equation $p_{n+1,0}\left(t\right)u_{0,2}\left(t\right)+p_{n+1,1}\left(t\right)u_{1,2}\left(t\right)+p_{n+1,2}\left(t\right)=g_{n+1,2}\left(t\right)$, which fixes $p_{n+1,2}\left(t\right)$ and so on. Thus, we obtain the remaining coefficients from the second set. This procedure can best be carried out by a computer if you want to see how the coefficients evolve. Note that the matrix $p_{-}\left(t\right)\mathrm{diag}\left(p_{ss}{\left(t\right)}^{-1}\right)$ also has the property that all its matrix coefficients are quotients of polynomial expressions in $\left\{g_{ij}\left(t\right)\right\}$. Now, we go back to the dressing matrices ${\mathbf{u}}_{-}(g,\gamma )$ and ${\mathbf{p}}_{-}(g,\gamma )$ from Theorem 1. Then, ${\mathbf{p}}_{-}(g,\gamma )=p_{-}{\left(t\right)}^{-1}$ and ${\mathbf{u}}_{-}(g,\gamma )={\left(p_{-}\left(t\right)\mathrm{diag}\left(p_{ss}{\left(t\right)}^{-1}\right)\right)}^{-1}$ and taking the inverse in both cases maintains the property that we want. So there holds

Theorem 2.

The matrix coefficients of the dressing matrices ${\mathbf{u}}_{-}(g,\gamma )$ and ${\mathbf{p}}_{-}(g,\gamma )$ from Theorem 1 are quotients of polynomial expressions in the analytic functions $\left\{g_{ij}\left(t\right)\right\}$.

Remark 3.

The result in Theorem 2 is for the $k\left[S\right]$-hierarchy and its strict version the analogue of the property that the dressing operators for KP and strict KP as constructed in [29,31] have coefficients that depend meromorphically on the group of commuting flows.

Let $\mathcal{K}\left(B\right)$ denote the subspace of compact operators in $\mathcal{L}\left(B\right)$. In the group $GL\left(B\right)$, we consider

$$G_{cpt}\left(B\right)=\{g\in GL\left(B\right)\mid u_{-}\left(g\right)\in \mathcal{K}\left(B\right)\}.$$

(44)

Clearly, ${\mathbf{P}}_{+}\left(B\right)$ is contained in $G_{cpt}\left(B\right)$. Note that for each $g\in G_{cpt}\left(B\right)$ the operator $p_{+}\left(g\right)=g-u_{-}\left(g\right)$ is a Fredholm operator of index zero. Using the fact that $\mathcal{K}\left(B\right)$ is a two-sided ideal in $\mathcal{L}\left(B\right)$ and some properties of Fredholm operators [44], one shows that $G_{cpt}\left(B\right)$ is a group. One can see $G_{cpt}\left(B\right)$ as the analogue for the $k\left[S\right]$-hierarchy and its strict version of the restricted linear group ${GL}_{res}\left(B\right)$ used to construct solutions of the KP hierarchy and its strict version in the Hilbert setting, see [29,31]. For there holds

Proposition 3.

The group $G_{cpt}\left(B\right)$ lies in the open set $\Gamma \Omega \left(B\right)$.

Proof. 

We prove the statement in the proposition by using the geometry of $G_{cpt}\left(B\right)$. Since B has the approximation property, the finite-dimensional operators $\mathcal{F}\left(B\right)$ in $\mathcal{L}\left(B\right)$ are dense in $\mathcal{K}\left(B\right)$. So it suffices to prove the statement for a $g\in G_{cpt}\left(B\right)$ with $u_{-}\left(g\right)\in \mathcal{F}\left(B\right).$ Let $B^{\ast }$ be the notation for the topological dual of B. Recall that the elements of $\mathcal{F}\left(B\right)$ are the image of the map $j:B\otimes B^{\ast }\to \mathcal{L}\left(B\right)$ defined by:

$$j(\sum _i{b}_i\otimes b_i^{\ast })\left(b\right)=\sum _i{b}_i^{\ast }\left(b\right)b_i,b_i^{\ast }\in B^{\ast },b\mathrm{and}\mathrm{the}\left\{b_i\right\}\mathit{all}\mathrm{in}B.$$

As vectors in B can be approximated arbitrarily close by vectors with a finite number of nonzero coordinates, we may assume that all the $\left\{b_i\right\}$ have that property and then we have reduced the claim to proving the statement for elements of $G_{cpt}\left(B\right)$ that decompose for some $N\in \mathbb{N}$ with respect to the splitting $B=B_{⩽N}\oplus B_{>N}$ as

$$\left[g\right]=\left(\begin{array}{cc}\left[g_N(1,1)\right]& \left[g_N(1,2)\right]\\ \left[0\right]& \left[g_N(2,2)\right]\end{array}\right),\mathrm{with}\left[g_N(1,1)\right]=\left(\begin{array}{ccc}{g}_{0,0}& \cdots & g_{0,N}\\ ⋮& \ddots & ⋮\\ g_{N,0}& \cdots & g_{N,N}\end{array}\right)\in {GL}_{N+1}\left(k\right)$$

$$\mathrm{and}\left[g_N(2,2)\right]=\left(\begin{array}{ccccc}{g}_{N+1,N+1}& \cdots & \cdots & g_{N,N+k}& \cdots \\ 0& \ddots & \cdots & ⋮& \cdots \\ ⋮& \ddots & \ddots & ⋮& \cdots \\ 0& \cdots & 0& g_{N+k,N+k}& \cdots \\ ⋮& ⋮& ⋮& \ddots & \ddots \end{array}\right)\mathrm{invertible}.$$

The matrix of each element $\gamma \in \Gamma$ we split similarly:

$$\left[\gamma \right]=\left(\begin{array}{cc}\left[{\gamma }_N(1,1)\right]& \left[{\gamma }_N(1,1)\right]\\ 0& \left[{\gamma }_N(2,2)\right]\end{array}\right),\mathrm{with}\left[{\gamma }_N(1,1)\right]=\left(\begin{array}{cccc}1& p_1\left(t\right)& \cdots & p_N\left(t\right)\\ 0& 1& \ddots & ⋮\\ ⋮& \ddots & \ddots & p_1\left(t\right)\\ 0& \cdots & 0& 1\end{array}\right)$$

Then, the matrix of the operator $\gamma g$ has the form

$$\begin{array}{c}\hfill \left[\gamma \right]\left[g\right]=\left(\begin{array}{cc}\left[{\gamma }_N(1,1)\right]\left[g_N(1,1)\right]& x(\gamma ,g)\\ 0& \left[{\gamma }_N(2,2)\right]\left[g_N(2,2)\right]\end{array}\right),\end{array}$$

where $x(\gamma ,g)=\left[{\gamma }_N(1,1)\right]\left[g_N(1,2)\right]+\left[\gamma (1,2)\right]\left[g_N(2,2)\right].$ Let ${Id}_{>N}$ be the matrix of the identity map in $GL\left(B_{>N}\right)$. Note that the map from $\Gamma \to k^N$ defined by

$$\gamma \left(t\right)\to {\overrightarrow{t}}_N=\{t_1,\cdots ,t_N\}$$

is a continuous surjection. Hence, if we find a nonzero open subset of $k^N$ such that for each vector ${\overrightarrow{t}}_N$ in that open set, it holds that

$$\left[{\gamma }_N(1,1)\right]\left[g_N(1,1)\right]=u\left({\overrightarrow{t}}_N\right)p\left({\overrightarrow{t}}_N\right)$$

with $p\left({\overrightarrow{t}}_N\right)$ as an invertible upper triangular $N+1\times N+1$-matrix and $u\left({\overrightarrow{t}}_N\right)$ a unipotent lower triangular matrix of the same size, then we have on an open subset of $\Gamma$

$$\left[\gamma \right]\left[g\right]=\left(\begin{array}{cc}u\left({\overrightarrow{t}}_N\right)& 0\\ 0& {Id}_{>N}\end{array}\right)\left(\begin{array}{cc}p\left({\overrightarrow{t}}_N\right)& x(\gamma ,g)\\ 0& \left[{\gamma }_N(2,2)\right]\left[g_N(2,2)\right]\end{array}\right)$$

and this is the decomposition we are looking for. Clearly, in order that we have the desired splitting of $\left[\gamma \right]\left[g\right]$, the condition

$$\left[{\gamma }_N(1,1)\right]\left[g_N(1,1)\right]=u\left({\overrightarrow{t}}_N\right)p\left({\overrightarrow{t}}_N\right)$$

for the vector ${\overrightarrow{t}}_N$ is also necessary. We will show by induction on N that there are N nonzero polynomials $\{r_1\left({\overrightarrow{t}}_N\right),\cdots ,r_N\left({\overrightarrow{t}}_N\right)\}$ such that for all ${\overrightarrow{t}}_N$ in the complement of the union of the zero sets of all the $\left\{r_i\right\}$ one has the decomposition $\left[{\gamma }_N(1,1)\right]\left[g_N(1,1)\right]=u\left({\overrightarrow{t}}_N\right)p\left({\overrightarrow{t}}_N\right)$. For $N=0$, the matrix $\left[g\right]$ is upper triangular and the desired decomposition holds for all ${\overrightarrow{t}}_N$. Now, we take $N⩾1$; we split off the first row of $\left[{\gamma }_N(1,1)\right]$ in $\left[{\gamma }_N(1,1)\right]\left[g_N(1,1)\right]$ as follows:

$$\left(\begin{array}{cc}1& 0\cdots \cdots 0\\ 0& \left[{\gamma }_{N-1}(1,1)\right]\end{array}\right)\left(\begin{array}{cc}1& p_1\left(t\right)\cdots p_N\left(t\right)\\ 0& {Id}_N\end{array}\right)\left(\begin{array}{ccc}{g}_{0,0}& \cdots & g_{0,N}\\ ⋮& \ddots & ⋮\\ g_{N,0}& \cdots & g_{N,N}\end{array}\right),$$

(45)

where ${Id}_N$ is the matrix of the identity map in ${GL}_N\left(k\right).$ We focus for the moment on the product of the last two matrices in (45). Since the first column of $\left[g_N(1,1)\right]$ is nonzero, the polynomial $r_N:=g_{0,0}+{\sum }_{k=1}^N{g}_{k,0}{p}_k\left(t\right)$ is nonzero. Now, we work on the complement of the zero set of $r_N$, so $r_N$ is invertible. Define for all $i,0⩽i⩽N,$ the polynomials ${\widehat{g}}_{0,i}=g_{0,i}+{\sum }_{k=1}^N{g}_{k,i}{p}_k\left(t\right)$. Note that ${\widehat{g}}_{0,0}=r_N.$ Then, the product of the last two matrices in (45) is equal to

$$\left(\begin{array}{cccc}{\widehat{g}}_{0,0}& \cdots & \cdots & {\widehat{g}}_{0,N}\\ g_{1,0}& g_{1,1}& \cdots & g_{1,N}\\ ⋮& ⋮& \ddots & ⋮\\ g_{N,0}& g_{N,1}& \cdots & g_{N,N}\end{array}\right)=\left(\begin{array}{ccccc}1& 0& \cdots & \cdots & 0\\ {\tilde{g}}_{1,0}& 1& 0& \cdots & \cdots \\ ⋮& 0& 1& 0& \cdots \\ ⋮& ⋮& \ddots & 1& 0\\ {\tilde{g}}_{N,0}& 0& \cdots & 0& 1\end{array}\right)\left(\begin{array}{cccc}{\widehat{g}}_{0,0}& \cdots & \cdots & {\widehat{g}}_{0,N}\\ 0& {\widehat{g}}_{1,1}& \cdots & {\widehat{g}}_{1,N}\\ ⋮& ⋮& \ddots & ⋮\\ 0& {\widehat{g}}_{N,1}& \cdots & {\widehat{g}}_{N,N}\end{array}\right),$$

where each ${\tilde{g}}_{k,0}=g_{k,0}{q}_N^{-1}$ and all ${\widehat{g}}_{ik}$ with $i⩾1$ and $k⩾1$ defined by ${\widehat{g}}_{ik}=g_{ik}-{\tilde{g}}_{i0}{\widehat{g}}_{0k}$. Next, we push the top row of the right matrix to the right

$$\left(\begin{array}{cccc}{\widehat{g}}_{0,0}& \cdots & \cdots & {\widehat{g}}_{0,N}\\ 0& {\widehat{g}}_{1,1}& \cdots & {\widehat{g}}_{1,N}\\ ⋮& ⋮& \ddots & ⋮\\ 0& {\widehat{g}}_{N,1}& \cdots & {\widehat{g}}_{N,N}\end{array}\right)=\left(\begin{array}{cccc}1& 0& \cdots & 0\\ 0& {\widehat{g}}_{1,1}& \cdots & {\widehat{g}}_{1,N}\\ ⋮& ⋮& \ddots & ⋮\\ 0& {\widehat{g}}_{N,1}& \cdots & {\widehat{g}}_{N,N}\end{array}\right)\left(\begin{array}{c}{\widehat{g}}_{0,0}\cdots {\widehat{g}}_{0,N}\\ 0{Id}_N\end{array}\right).$$

The matrix $\left(\begin{array}{c}{\widehat{g}}_{0,0}\cdots {\widehat{g}}_{0,N}\\ 0{Id}_N\end{array}\right)$ at the right has determinant ${\widehat{g}}_{0,0}=r_N\ne 0$ and will be part of $p\left({\overrightarrow{t}}_N\right)$. Next, we move the matrix $\left(\begin{array}{cc}1& 0\cdots \cdots 0\\ 0& \left[{\gamma }_{N-1}(1,1)\right]\end{array}\right)$ in product (45) to the right by using

$$\left(\begin{array}{cc}1& 0\cdots \cdots 0\\ 0& \left[{\gamma }_{N-1}(1,1)\right]\end{array}\right)\left(\begin{array}{ccccc}1& 0& \cdots & \cdots & 0\\ {\tilde{g}}_{1,0}& 1& 0& \cdots & \cdots \\ ⋮& 0& 1& 0& \cdots \\ ⋮& ⋮& \ddots & 1& 0\\ {\tilde{g}}_{N,0}& 0& \cdots & 0& 1\end{array}\right)=\left(\begin{array}{cc}1& 0\\ \overrightarrow{z}& {Id}_N\end{array}\right)\left(\begin{array}{cc}1& 0\cdots \cdots 0\\ 0& \left[{\gamma }_{N-1}(1,1)\right]\end{array}\right),$$

where $\overrightarrow{z}$ is the column of length N equal to $\left[{\gamma }_{N-1}(1,1)\right]\overrightarrow{\tilde{g}}$ with ${\left(\overrightarrow{\tilde{g}}\right)}^T=({\tilde{g}}_{1,0},\cdots ,{\tilde{g}}_{N,0})$. The matrix $\left(\begin{array}{cc}1& 0\\ \overrightarrow{z}& {Id}_N\end{array}\right)$ will be part of $u\left({\overrightarrow{t}}_N\right)$. Thus, we reduce the case to the product

$$\left(\begin{array}{cc}1& 0\cdots \cdots 0\\ 0& \left[{\gamma }_{N-1}(1,1)\right]\end{array}\right)\left(\begin{array}{cccc}1& 0& \cdots & 0\\ 0& {\widehat{g}}_{1,1}& \cdots & {\widehat{g}}_{1,N}\\ ⋮& ⋮& \ddots & ⋮\\ 0& {\widehat{g}}_{N,1}& \cdots & {\widehat{g}}_{N,N}\end{array}\right),$$

where the matrix at the right has determinant $r_N^{-1}\ne 0$. The induction hypothesis gives us then the sequence of nonzero polynomials $\{r_1,\cdots ,r_N\}$, so that on the complement of the union of all their zeros we have the desired decomposition. This proves the claim in the proposition. □

From Theorem 1 and Proposition 3, we may conclude

Corollary 2.

The manifold $G_{cpt}\left(B\right)/{\mathbf{P}}_{+}\left(H\right)$ describes solutions of the $k\left[S\right]$-hierarchy and the manifold $G_{cpt}\left(B\right)/{\mathbf{U}}_{+}\left(H\right)$ describes solutions of the strict $k\left[S\right]$-hierarchy. As such, these manifolds are in the present context the analogues of the Grassmann manifold Gr(B) and its cover, the flag variety ${\mathcal{F}}_1$, used in [29,31] to construct solutions of the KP hierarchy and the strict KP hierarchy.

6. Conclusions

The achievements of the present paper are threefold: first of all, we constructed a far larger collection of dressing matrices for both the $k\left[S\right]$-hierarchy and its strict version. Secondly, we showed that the form of the matrix coefficients of both sets of dressing matrices is reminiscent of that of the coefficients of the wave functions of the KP hierarchy and its strict version. Finally, by the introduction of the group $G_{cpt}\left(B\right)$ and its property Proposition 3, we created a Segal–Wilson-type description of both sets of solutions from $G_{cpt}\left(B\right)$ as homogeneous spaces of this group.

We mention a number of research topics for future research. Now, that we are less restricted by the Hilbert–Schmidt condition in [41], it makes sense to study the reductions in the $k\left[S\right]$-hierarchy and its strict version and to give a description of their solutions in $\Gamma \Omega \left(S\right)$. They should be analogues of the descriptions given in [27,29,45].

A second theme could be to find expressions in Fredholm determinants for the coefficients of the dressing matrices that we have found. They should be analogues of expressions found in [29,32] for the wave functions for the KP hierarchy and its strict version and it would be interesting to find out which choices of $g\in G_{cpt}\left(B\right)$ lead to special functions.

A third topic would be to see if the present hierarchies possess solutions that correspond to algebraic geometrical data, like the solutions for the KP hierarchy and the Gelfand–Dickey hierarchies as found by Krichever in [16,17]. They have been shown to fit nicely into the geometric description from [29] and also in this situation we hope to benefit our wider class of dressing matrices.

A final theme, would be to find proper instances in the theoretical physics literature, where the $k\left[S\right]$-hierarchy or its strict version plays a role.