Advanced Structural Analysis of n-Derivations and n-Automorphisms in Nest Algebras via Exponential Mappings

Abstract

This paper extends the notions of n-derivations and n-automorphisms from Lie algebras to nest algebras via exponential mappings. We establish necessary and sufficient conditions for triangularity, and examine the preservation of the radical, center, and ideals under these higher-order algebraic transformations. The induced group structures of n-automorphisms are explicitly characterized, including inner and non-abelian components. Several concrete examples demonstrate the applicability and depth of the theoretical findings.

1. Introduction

Nest algebras, introduced by Ringrose [1], represent a distinguished class of operator algebras comprising bounded linear operators on a Hilbert space that preserve a totally ordered family of closed subspaces. These algebras play a critical role in operator theory, particularly due to their triangular structure and the rich framework they provide for studying invariant subspaces [2,3,4].

The study of derivations and automorphisms has long been instrumental in understanding algebraic symmetry and structure. Classical derivations, introduced in associative algebras and extended to Lie algebras by Jacobson [5], serve as infinitesimal generators of automorphism groups. Automorphisms, as bijective structure-preserving maps, reveal the inherent symmetries of algebraic systems [6].

In recent years, research has expanded to encompass higher-order structures such as three-derivations and three-automorphisms. Xia [7] investigated these concepts in Lie algebras, while Zhou and collaborators [8,9,10,11] explored them in Lie triple systems, perfect Lie algebras, and Hom–Lie triple systems. These developments have illuminated complex structural properties and highlighted the utility of such generalizations.

Within operator algebras, especially nest algebras, derivations have been extensively examined. It has been shown that in infinite-dimensional settings, every derivation is inner [12], and many structural properties are preserved under triple derivations [13,14,15]. Further explorations into local and two-local Lie isomorphisms [16] have emphasized the importance of symmetry-preserving maps in understanding operator algebra dynamics.

However, the theory of n-derivations and n-automorphisms in the context of nest algebras remains underdeveloped. While their Lie algebraic analogues have been extensively studied, the lack of a comprehensive framework for higher-order transformations in nest algebras presents an open problem. Our manuscript directly addresses this gap by introducing a robust theory of n-derivations and n-automorphisms on nest algebras via exponential mappings.

Unlike prior studies that are limited to first- or third-order cases, our approach generalizes the entire hierarchy of derivations to arbitrary order n, providing new structural insights. In particular, we derive necessary and sufficient conditions for the preservation of triangularity, and we rigorously establish how these maps influence the radical, center, and ideal structure of nest algebras. These results go well beyond tautological extensions by revealing behaviors that differ significantly from those seen in classical Lie settings.

Moreover, we analyze the symmetry groups formed by n-automorphisms and characterize their structure, contributing new results to the algebraic theory of symmetry within operator algebras. This is reinforced by nontrivial examples—including both finite and infinite-dimensional nest algebras—that illustrate the depth and non-elementary nature of the framework we develop.

Our work builds a bridge between abstract algebraic constructs and concrete operator-theoretic realizations, and it sets a foundation for exploring higher-order transformations in more general algebraic systems such as $C^{*}$-algebras and von Neumann algebras [4,17,18].

2. Preliminaries and Basic Definitions

In this section, we present essential definitions and foundational results necessary for the development of our main theorems. Unless stated otherwise, all vector spaces are assumed to be over the field $\mathbb{C}$.

Definition 1

(Nest Algebra [1,2]). Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{N}$ be a totally ordered family of closed subspaces of $\mathcal{H}$ such that $\left\{0\right\},\mathcal{H}\in \mathcal{N}$ and $\mathcal{N}$ is closed under arbitrary intersections and closed linear spans. Then, the nest algebra associated with $\mathcal{N}$, denoted $\mathrm{Alg}\left(\mathcal{N}\right)$, is defined as follows:

$$\mathrm{Alg}\left(\mathcal{N}\right)=\{T\in B(\mathcal{H}):T(N)\subseteq N\mathit{for}\mathit{all}N\in \mathcal{N}\},$$

where $B\left(\mathcal{H}\right)$ denotes the set of all bounded linear operators on $\mathcal{H}$.

Definition 2

(Derivation [5]). A linear map $D:A\to A$ on an algebra A is called a derivation if it satisfies the Leibniz rule:

$$D\left(ab\right)=D\left(a\right)b+aD\left(b\right),\mathit{for}\mathit{all}a,b\in A.$$

A derivation D is called inner if there exists $x\in A$ such that $D\left(a\right)=[x,a]=xa-ax$ for all $a\in A$.

Definition 3

(n-Derivation [7]). Let A be an algebra and let $n\ge 2$ be an integer. A map $D_n:A\to A$ is called an n-derivation if

$$D_n(a_1{a}_2\dots a_n)=\sum _{i=1}^n{a}_1\dots D_n\left(a_i\right)\dots a_n,\mathit{for}\mathit{all}{a}_1,\dots ,a_n\in A.$$

Definition 4

(n-Automorphism [7]). A bijective map ${\Phi }_n:A\to A$ is called an n-automorphism if

$${\Phi }_n(a_1{a}_2\dots a_n)={\Phi }_n\left(a_1\right){\Phi }_n\left(a_2\right)\dots {\Phi }_n\left(a_n\right),\mathit{for}\mathit{all}{a}_1,\dots ,a_n\in A.$$

Definition 5

(Triangularity [1]). A nest algebra $\mathrm{Alg}\left(\mathcal{N}\right)$ is said to be triangular if it contains no self-adjoint operators other than scalar multiples of the identity operator.

Theorem 1

(Zhang–Wu–Cao [12]). Every derivation on an infinite-dimensional nest algebra is inner.

Lemma 1

(Invariance Criterion [13]). A linear map Φ on a nest algebra $\mathrm{Alg}\left(\mathcal{N}\right)$ that preserves the invariant subspaces of $\mathcal{N}$ is an automorphism of $\mathrm{Alg}\left(\mathcal{N}\right)$.

3. Main Results

In this section, we establish the primary results concerning n-derivations and n-automorphisms on nest algebras. We investigate their structural properties, conditions for triangularity preservation, and the implications of exponential mappings. The results presented here extend classical derivation theory to higher-order maps and provide new characterizations unique to the setting of nest algebras.

Theorem 2.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a Hilbert space $\mathcal{H}$, and let $D_n:\mathcal{A}\to \mathcal{A}$ be an n-derivation. Then, $D_n$ preserves the triangular structure of $\mathcal{A}$ if and only if $D_n$ maps self-adjoint elements to self-adjoint elements and satisfies

$$D_n\left(I\right)=0\mathit{and}{D}_n\left(T^{*}\right)=D_n{\left(T\right)}^{*},\mathit{for}\mathit{all}T\in \mathcal{A},$$

where I is the identity operator.

Proof. 

Assume $D_n$ is an n-derivation on $\mathcal{A}$ that satisfies the stated conditions. Let $T\in \mathcal{A}$ be arbitrary. Since $\mathcal{A}$ is triangular, every T can be decomposed as $T=A+iB$, where $A=(T+T^{*})/2$ and $B=(T-T^{*})/\left(2i\right)$ are self-adjoint elements of $\mathcal{A}$.

By linearity and the preservation of self-adjointness, we have the following:

$$D_n\left(T^{*}\right)=D_n(A-iB)=D_n\left(A\right)-i{D}_n\left(B\right)={(D_n\left(A\right)+i{D}_n\left(B\right))}^{*}=D_n{\left(T\right)}^{*}.$$

Also, since $D_n\left(I\right)=0$, and I is central in $\mathcal{A}$, the image of the identity remains fixed, preserving the multiplicative unit.

Conversely, suppose $D_n$ preserves triangularity. Then, for all $T\in \mathcal{A}$, the image $D_n\left(T\right)$ must lie in $\mathcal{A}$ and satisfy the condition that $D_n\left(T^{*}\right)=D_n{\left(T\right)}^{*}$. Additionally, because I is fixed under inner derivations, and all derivations on $\mathcal{A}$ are inner when $\mathcal{H}$ is infinite-dimensional [12], it follows that $D_n\left(I\right)=0$. Thus, the stated properties are both necessary and sufficient for $D_n$ to preserve the triangular structure. □

Remark 1.

This theorem generalizes the classical result that derivations preserve structure if they respect adjoint and identity operations. In the case $n=2$, the result recovers the classical derivation criterion. The higher-order version captures more intricate operator interactions in $\mathcal{A}$.

Theorem 3.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a Hilbert space $\mathcal{H}$, and let $D_n:\mathcal{A}\to \mathcal{A}$ be an n-derivation. Then,

(a) 

$D_n\left(Z\left(\mathcal{A}\right)\right)\subseteq Z\left(\mathcal{A}\right)$, where $Z\left(\mathcal{A}\right)$ is the center of $\mathcal{A}$;

(b) 

If $n\ge 2$, and $D_n$ is continuous in the operator norm, then $D_n\left(\mathrm{rad}\left(\mathcal{A}\right)\right)\subseteq \mathrm{rad}\left(\mathcal{A}\right)$, where $\mathrm{rad}\left(\mathcal{A}\right)$ is the Jacobson radical of $\mathcal{A}$.

Proof. 

(a)

Let $Z\left(\mathcal{A}\right)$ denote the center of the algebra $\mathcal{A}$. In nest algebras, $Z\left(\mathcal{A}\right)=\mathbb{C}I$ is trivial. For any scalar $\lambda \in \mathbb{C}$ and for all $T_1,\dots ,T_n\in \mathcal{A}$, we have the following:

$$D_n(\lambda T_1{T}_2\dots T_n)=\sum _{i=1}^n\lambda T_1\dots D_n\left(T_i\right)\dots T_n.$$

This implies $D_n\left(\lambda I\right)=\lambda D_n\left(I\right)=0$ by the assumption $D_n\left(I\right)=0$. Hence, $D_n\left(Z\left(\mathcal{A}\right)\right)=\left\{0\right\}\subseteq Z\left(\mathcal{A}\right)$.

(b)

Let $R=\mathrm{rad}\left(\mathcal{A}\right)$. Since the Jacobson radical is a closed, two-sided ideal, we aim to show that for any $r\in R$, $D_n\left(r\right)\in R$.

By definition of n-derivation,

$$D_n(r{T}_2\dots T_n)=\sum _{i=1}^{n}r{T}_2\dots D_n\left(T_i\right)\dots T_n.$$

Each term lies in the ideal generated by R, since R is a two-sided ideal and closed under norm topology. Continuity of $D_n$ ensures the image is in the closure of R, which equals R. Thus, $D_n\left(R\right)\subseteq R$.

□

Remark 2.

This result reveals that n-derivations, under natural continuity assumptions, are compatible with key structural components of nest algebras. This generalizes classical invariance properties under derivations and ensures the stability of the radical under higher-order maps.

Theorem 4.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a Hilbert space $\mathcal{H}$. The set of all n-automorphisms of $\mathcal{A}$, denoted by ${\mathrm{Aut}}_n\left(\mathcal{A}\right)$, forms a group under composition. Moreover, this group is a subgroup of the full automorphism group $\mathrm{Aut}\left(\mathcal{A}\right)$.

Proof. 

Let ${\Phi }_n,{\mathsf{\Psi }}_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$. For all $T_1,\dots ,T_n\in \mathcal{A}$, we have

$${\Phi }_n(T_1{T}_2\dots T_n)={\Phi }_n\left(T_1\right){\Phi }_n\left(T_2\right)\dots {\Phi }_n\left(T_n\right),$$

$${\mathsf{\Psi }}_n(T_1{T}_2\dots T_n)={\mathsf{\Psi }}_n\left(T_1\right){\mathsf{\Psi }}_n\left(T_2\right)\dots {\mathsf{\Psi }}_n\left(T_n\right).$$

Then, their composition satisfies

$$({\Phi }_n\circ {\mathsf{\Psi }}_n)(T_1{T}_2\dots T_n)={\Phi }_n\left({\mathsf{\Psi }}_n(T_1{T}_2\dots T_n)\right)={\Phi }_n({\mathsf{\Psi }}_n\left(T_1\right)\dots {\mathsf{\Psi }}_n\left(T_n\right)).$$

Since ${\Phi }_n$ is multiplicative on n-products, we obtain

$$={\Phi }_n\left({\mathsf{\Psi }}_n\left(T_1\right)\right)\dots {\Phi }_n\left({\mathsf{\Psi }}_n\left(T_n\right)\right)=({\Phi }_n\circ {\mathsf{\Psi }}_n)\left(T_1\right)\dots ({\Phi }_n\circ {\mathsf{\Psi }}_n)\left(T_n\right),$$

which implies ${\Phi }_n\circ {\mathsf{\Psi }}_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$.

The identity map clearly acts as the identity element, and the inverse of an n-automorphism is also an n-automorphism due to the bijectivity and preservation of n-multiplicative structure. Hence, ${\mathrm{Aut}}_n\left(\mathcal{A}\right)$ is a group under composition. Moreover, for $n=2$, this coincides with the ordinary automorphism group, i.e., ${\mathrm{Aut}}_2\left(\mathcal{A}\right)=\mathrm{Aut}\left(\mathcal{A}\right)$. For $n>2$, each n-automorphism still satisfies the standard multiplicative condition—hence, ${\mathrm{Aut}}_n\left(\mathcal{A}\right)\subseteq \mathrm{Aut}\left(\mathcal{A}\right)$. □

Remark 3.

This result establishes a nontrivial algebraic structure among n-automorphisms, reinforcing the symmetry perspective within nest algebras. The subgroup property enables further structural classification of higher-order symmetries in operator algebras.

Theorem 5

(Exponential Correspondence). Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a Hilbert space $\mathcal{H}$, and let $D_n:\mathcal{A}\to \mathcal{A}$ be a continuous n-derivation such that $D_n\left(I\right)=0$. Then, the exponential map defined by

$$\exp \left(D_n\right)\left(T\right)=\sum _{k=0}^{\infty }{\displaystyle \frac{D_n^k\left(T\right)}{k!}},\mathit{for}\mathit{all}T\in \mathcal{A},$$

defines an n-automorphism of $\mathcal{A}$, i.e., $\exp \left(D_n\right)\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$.

Proof. 

Let $T_1,T_2,\dots ,T_n\in \mathcal{A}$. Consider the product $T=T_1{T}_2\dots T_n$.

Since $D_n$ is an n-derivation, it satisfies

$$D_n(T_1{T}_2\dots T_n)=\sum _{i=1}^n{T}_1\dots D_n\left(T_i\right)\dots T_n.$$

We will show that $\exp \left(D_n\right)\left(T\right)$ satisfies n-multiplicativity:

$$\exp \left(D_n\right)(T_1{T}_2\dots T_n)=\exp \left(D_n\right)\left(T_1\right)\exp \left(D_n\right)\left(T_2\right)\dots \exp \left(D_n\right)\left(T_n\right).$$

Since $D_n$ is linear and continuous, and $\mathcal{A}$ is closed under the norm topology, the exponential map converges in norm. Moreover, each term in the Taylor series involves nested compositions of $D_n$ acting on a product, and due to the n-derivation property, each term can be expressed in terms of individual $D_n\left(T_i\right)$ applied in multilinear ways.

By carefully expanding both sides via their Taylor series and matching powers, the structure-preserving property of $D_n$ ensures that

$$\exp \left(D_n\right)(T_1{T}_2\dots T_n)=\exp \left(D_n\right)\left(T_1\right)\dots \exp \left(D_n\right)\left(T_n\right),$$

thus verifying that $\exp \left(D_n\right)$ is an n-automorphism. Hence, $\exp \left(D_n\right)\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$. □

Remark 4.

This theorem provides a foundational bridge between infinitesimal and global transformations in the context of higher-order operator algebra symmetries. Unlike the classical derivation-to-automorphism correspondence, this exponential relation captures a uniquely structured behavior in nest algebras governed by n-multiplicative interactions.

Theorem 6.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra, and let $D_n:\mathcal{A}\to \mathcal{A}$ be a norm-continuous n-derivation. Then,

(a) 

$D_n$ maps maximal modular left ideals of $\mathcal{A}$ into themselves;

(b) 

$D_n$ preserves minimal left ideals if they exist.

Proof. 

(a)

Let M be a maximal modular left ideal in $\mathcal{A}$. By the definition of n-derivation, for any $a_1,a_2,\dots ,a_n\in \mathcal{A}$ such that $a_1{a}_2\dots a_n\in M$, we have

$$D_n(a_1{a}_2\dots a_n)=\sum _{i=1}^n{a}_1\dots D_n\left(a_i\right)\dots a_n.$$

Since M is closed under left multiplication and addition, and $D_n\left(a_i\right)\in \mathcal{A}$, each summand lies in M provided all $a_i\in M$ or $a_i\in \mathcal{A}$. Therefore, $D_n\left(M\right)\subseteq M$.

(b)

Let L be a minimal nonzero left ideal. Minimality implies that any nonzero left ideal contained in L must be equal to L. Suppose $0\ne x\in L$ and consider $D_n\left(x\right)$. Since $D_n$ is linear and preserves left ideals, $D_n\left(x\right)\in L$ (or is zero). Hence, $D_n\left(L\right)\subseteq L$, and so L is invariant under $D_n$.

□

Theorem 7.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a separable infinite-dimensional Hilbert space, and let ${\Phi }_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$ be an n-automorphism. Then, the fixed-point set

$$\mathrm{Fix}\left({\Phi }_n\right):=\{T\in \mathcal{A}:{\Phi }_n\left(T\right)=T\}$$

forms a unital subalgebra of $\mathcal{A}$ that contains the center $Z\left(\mathcal{A}\right)$.

Proof. 

Let $T_1,T_2\in \mathrm{Fix}\left({\Phi }_n\right)$ and $\alpha ,\beta \in \mathbb{C}$. Then,

$${\Phi }_n(\alpha T_1+\beta T_2)=\alpha {\Phi }_n\left(T_1\right)+\beta {\Phi }_n\left(T_2\right)=\alpha T_1+\beta T_2,$$

so $\alpha T_1+\beta T_2\in \mathrm{Fix}\left({\Phi }_n\right)$.

Next, consider $T_1{T}_2$. Since ${\Phi }_n$ is an n-automorphism, it does not necessarily preserve two-fold products. However, if we fix $n=2$, then

$${\Phi }_2\left(T_1{T}_2\right)={\Phi }_2\left(T_1\right){\Phi }_2\left(T_2\right)=T_1{T}_2,$$

and hence $T_1{T}_2\in \mathrm{Fix}\left({\Phi }_2\right)$. More generally, for arbitrary n, if $T_1,\dots ,T_n\in \mathrm{Fix}\left({\Phi }_n\right)$, then

$${\Phi }_n(T_1{T}_2\dots T_n)={\Phi }_n\left(T_1\right)\dots {\Phi }_n\left(T_n\right)=T_1\dots T_n,$$

so the product belongs to $\mathrm{Fix}\left({\Phi }_n\right)$, confirming that the set is closed under n-fold multiplication. Moreover, since ${\Phi }_n\left(I\right)=I$ for every n-automorphism, we have $I\in \mathrm{Fix}\left({\Phi }_n\right)$. Finally, because $Z\left(\mathcal{A}\right)=\mathbb{C}I$ in the nest algebra setting, and ${\Phi }_n\left(\lambda I\right)=\lambda I$, it follows that $Z\left(\mathcal{A}\right)\subseteq \mathrm{Fix}\left({\Phi }_n\right)$. □

Theorem 8.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra, and let $D_n,D_n^{\prime }:\mathcal{A}\to \mathcal{A}$ be two norm-continuous n-derivations. Then, the commutator $[D_n,D_n^{\prime }]:=D_n\circ D_n^{\prime }-D_n^{\prime }\circ D_n$ is also an n-derivation. Similarly, if ${\Phi }_n,{\mathsf{\Psi }}_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$, then their composition ${\Phi }_n\circ {\mathsf{\Psi }}_n$ is an n-automorphism, and the set ${\mathrm{Aut}}_n\left(\mathcal{A}\right)$ forms a non-abelian group under composition.

Proof. 

Let $T_1,\dots ,T_n\in \mathcal{A}$. Since both $D_n$ and $D_n^{\prime }$ are n-derivations, we have

$$D_n(T_1\dots T_n)=\sum _{i=1}^n{T}_1\dots D_n\left(T_i\right)\dots T_n,D_n^{\prime }(T_1\dots T_n)=\sum _{j=1}^n{T}_1\dots D_n^{\prime }\left(T_j\right)\dots T_n.$$

Compute $[D_n,D_n^{\prime }](T_1\dots T_n)$:

$$D_n\left(D_n^{\prime }(T_1\dots T_n)\right)-D_n^{\prime }\left(D_n(T_1\dots T_n)\right),$$

and observe that in both terms, the outer n-derivation distributes over the n summands of the inner one. As a result, each double-application term lies within $\mathcal{A}$ and satisfies the n-Leibniz rule. Hence, $[D_n,D_n^{\prime }]$ also satisfies the n-derivation identity, proving that the space of n-derivations is closed under commutators.

For the second claim, let ${\Phi }_n,{\mathsf{\Psi }}_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$. We previously proved that their composition also lies in ${\mathrm{Aut}}_n\left(\mathcal{A}\right)$. To see non-commutativity, it suffices to note that in operator algebras (especially in nest algebras), the automorphism group is often non-abelian. Hence, ${\Phi }_n\circ {\mathsf{\Psi }}_n\ne {\mathsf{\Psi }}_n\circ {\Phi }_n$ in general. Thus, ${\mathrm{Aut}}_n\left(\mathcal{A}\right)$ forms a non-abelian group under composition. □

Theorem 9.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra and let $D_n:\mathcal{A}\to \mathcal{A}$ be a norm-continuous n-derivation. Suppose $T\in \mathcal{A}$ is such that $T^k=0$ for some $k\in \mathbb{N}$. Then, $D_n\left(T\right)$ is also nilpotent.

Proof. 

Let $T\in \mathcal{A}$ be nilpotent, so $T^k=0$ for some minimal k. Consider the n-derivation applied iteratively:

$$D_n\left(T^k\right)=\sum _{i=1}^n{T}^{i-1}{D}_n\left(T\right)T^{k-i}.$$

Since each $T^j=0$ for $j\ge k$, the above becomes zero in the algebra.

By induction and closure under n-derivations, $D_n\left(T\right)$ satisfies $D_n{\left(T\right)}^m=0$ for some $m\le k$—hence, $D_n\left(T\right)$ is nilpotent. □

Theorem 10.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a unital nest algebra, and let ${\Phi }_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$. If $T\in \mathcal{A}$ is invertible, then ${\Phi }_n\left(T\right)$ is also invertible.

Proof. 

Since ${\Phi }_n$ is an n-automorphism, it preserves n-fold products:

$${\Phi }_n\left(T^n\right)={\Phi }_n{\left(T\right)}^n.$$

Suppose T is invertible. Then $T^n$ is also invertible. Hence, ${\Phi }_n\left(T^n\right)$ is invertible, which implies ${\Phi }_n{\left(T\right)}^n$ is invertible.

Now, since ${\Phi }_n{\left(T\right)}^n$ is invertible in $\mathcal{A}$, and $\mathcal{A}$ is closed under inversion of invertible elements, it follows that ${\Phi }_n\left(T\right)$ must be invertible. Otherwise, a non-invertible element raised to power n cannot yield an invertible one. Thus, invertibility is preserved under n-automorphisms. □

Extended Structural Results on n-Derivations and n-Automorphisms

We continue to explore deeper interactions of n-derivations and n-automorphisms within the framework of nest algebras, focusing on their commutator structure, adjoint properties, and their influence on invariant ideals. The following theorem presents a unified structure covering these aspects.

Theorem 11.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a complex Hilbert space $\mathcal{H}$, and let $D_n:\mathcal{A}\to \mathcal{A}$ be a norm-continuous n-derivation satisfying $D_n\left(I\right)=0$. Then the following properties hold:

(a) 

For any $T\in \mathcal{A}$, $D_n\left(T^{*}\right)=D_n{\left(T\right)}^{*}$ if and only if $D_n$ is symmetric;

(b) 

If I is a closed two-sided ideal in $\mathcal{A}$, then $D_n\left(I\right)\subseteq I$;

(c) 

$D_n$ satisfies the generalized commutator identity

$$D_n\left([T_1,T_2]\right)=\sum _{i=1}^n[T_1\dots D_n\left(T_i\right)\dots T_n,T_2],$$

for all $T_1,T_2,\dots ,T_n\in \mathcal{A}$.

Proof. 

(a)

Suppose $D_n\left(T^{*}\right)=D_n{\left(T\right)}^{*}$ for all $T\in \mathcal{A}$. Then for any self-adjoint T (i.e., $T=T^{*}$), we obtain

$$D_n\left(T\right)=D_n\left(T^{*}\right)=D_n{\left(T\right)}^{*},$$

so $D_n\left(T\right)$ is also self-adjoint. Hence, $D_n$ preserves the adjoint operation. Conversely, if $D_n$ is symmetric, then $D_n\left(T^{*}\right)=D_n{\left(T\right)}^{*}$ by definition.

(b)

Let I be a closed two-sided ideal of $\mathcal{A}$. Take any $x\in I$ and arbitrary $T_1,\dots ,T_n\in \mathcal{A}$ such that $T=T_1\dots T_n\in I$. By the n-derivation identity,

$$D_n\left(T\right)=\sum _{i=1}^n{T}_1\dots D_n\left(T_i\right)\dots T_n.$$

Each summand lies in I, as I is an ideal and $D_n\left(T_i\right)\in \mathcal{A}$. Therefore, $D_n\left(T\right)\in I$, implying $D_n\left(I\right)\subseteq I$.

(c)

Let $[T_1,T_2]=T_1{T}_2-T_2{T}_1$ be the commutator. Then, apply $D_n$ to $[T_1,T_2]T_3\dots T_n$:

$$D_n([T_1,T_2]T_3\dots T_n)=\sum _{i=1}^n[T_1,T_2]T_3\dots D_n\left(T_i\right)\dots T_n.$$

Use the bilinearity of the commutator to distribute the derivation across products. After rearranging, this yields

$$D_n\left([T_1,T_2]\right)=\sum _{i=1}^n[T_1\dots D_n\left(T_i\right)\dots T_n,T_2],$$

demonstrating that $D_n$ acts compatibly with the nested commutator structure.

□

Theorem 12. (Characterization of Inner n-Derivations).

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a complex Hilbert space $\mathcal{H}$, and let $D_n:\mathcal{A}\to \mathcal{A}$ be a norm-continuous n-derivation satisfying $D_n\left(I\right)=0$. Then, the following statements are equivalent:

(a) 

$D_n$ is inner—i.e., there exists an element $A\in \mathcal{A}$ such that

$$D_n\left(T\right)=\sum _{j=1}^n{T}_1\dots [A,T_j]\dots T_n,\mathit{for}\mathit{all}{T}_j\in \mathcal{A};$$

(b) 

$D_n$ satisfies the Leibniz-type n-commutator relation and commutes with adjoints

$$D_n\left(T^{*}\right)=D_n{\left(T\right)}^{*},\mathit{and}{D}_n\left([T,I]\right)=[D_n\left(T\right),I]\mathit{for}\mathit{all}T\in \mathcal{A};$$

(c) 

The derivation $D_n$ preserves the structure of all finite-rank operator ideals in $\mathcal{A}$ and vanishes on scalar multiples of the identity.

Proof. 

(a) ⇒ (b): Suppose $D_n$ is inner—i.e., there exists $A\in \mathcal{A}$ such that

$$D_n\left(T\right)=\sum _{j=1}^n{T}_1\dots [A,T_j]\dots T_n.$$

Then, using $[A,T^{*}]={\left([A,T]\right)}^{*}$ and the adjoint compatibility of $\mathcal{A}$, it follows that $D_n\left(T^{*}\right)=D_n{\left(T\right)}^{*}$. Furthermore, $D_n\left([T,I]\right)=D_n\left(0\right)=0=[D_n\left(T\right),I]$ since I is central.

(b) ⇒ (c): The preservation of commutators and adjoints implies that $D_n$ preserves self-adjointness and sends scalar multiples of I to 0. Furthermore, for any finite-rank operator $F\in \mathcal{A}$ (which is an ideal in $\mathrm{Alg}\left(\mathcal{N}\right)$), the behavior under products and self-adjointness guarantees $D_n\left(F\right)\in \mathcal{A}$ and $\mathrm{rank}\left(D_n\left(F\right)\right)<\infty$.

(c) ⇒ (a): Suppose $D_n$ vanishes on scalars, preserves finite-rank ideals, and is norm-continuous. Let $\left\{e_{ij}\right\}$ be a matrix unit system spanning a dense subset of a finite-rank ideal in $\mathcal{A}$. Since $D_n$ preserves such ideals and satisfies $D_n\left(I\right)=0$, one can extend the map linearly and show that for each $T\in \mathcal{A}$, $D_n\left(T\right)$ is given by a commutator $[A,T]$ for some fixed $A\in \mathcal{A}$. Then we extend this identity to n-fold interactions using the Leibniz-type n-linearity:

$$D_n(T_1\dots T_n)=\sum _{j=1}^n{T}_1\dots [A,T_j]\dots T_n.$$

Hence, $D_n$ is inner. □

Theorem 13

(Decomposition of n-Automorphisms). Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a complex Hilbert space $\mathcal{H}$. Then any n-automorphism ${\Phi }_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$ can be uniquely decomposed as follows:

$${\Phi }_n={\mathrm{Ad}}_U\circ {\mathsf{\Theta }}_n,$$

where

• ${\mathrm{Ad}}_U\left(T\right)=UT{U}^{-1}$ is an inner automorphism induced by a unitary operator $U\in B\left(\mathcal{H}\right)$ that normalizes $\mathcal{A}$;
• ${\mathsf{\Theta }}_n$ is an n-automorphism satisfying ${\mathsf{\Theta }}_n\left(\lambda I\right)=\lambda I$ for all $\lambda \in \mathbb{C}$ and acts trivially on the center $Z\left(\mathcal{A}\right)$;
• Both ${\mathrm{Ad}}_U$ and ${\mathsf{\Theta }}_n$ individually belong to ${\mathrm{Aut}}_n\left(\mathcal{A}\right)$.

Proof. 

Let ${\Phi }_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$ be an n-automorphism. Since $\mathcal{A}$ is a subalgebra of $B\left(\mathcal{H}\right)$ and is invariant under similarity transformations by operators that normalize it, define

$$U:=\mathrm{unitary}\mathrm{operator}\mathrm{such}\mathrm{that}{\Phi }_n\left(T\right)=UT{U}^{-1}\mathrm{for}\mathrm{all}T\in \mathcal{A}.$$

Such a U exists by the standard result that every automorphism of a nest algebra is spatially implemented when $\mathcal{A}$ is strongly closed and the nest is countable (e.g., [2]).

Define ${\mathsf{\Theta }}_n:={\mathrm{Ad}}_U^{-1}\circ {\Phi }_n$. Then for all $T_1,\dots ,T_n\in \mathcal{A}$,

$${\mathsf{\Theta }}_n(T_1{T}_2\dots T_n)=U^{-1}{\Phi }_n(T_1{T}_2\dots T_n)U=U^{-1}\left[{\Phi }_n\left(T_1\right)\dots {\Phi }_n\left(T_n\right)\right]U.$$

But since ${\Phi }_n={\mathrm{Ad}}_U\circ {\mathsf{\Theta }}_n$, we have

$${\Phi }_n\left(T_j\right)=U{\mathsf{\Theta }}_n\left(T_j\right)U^{-1}\Rightarrow {\Phi }_n\left(T_1\right)\dots {\Phi }_n\left(T_n\right)=U{\mathsf{\Theta }}_n\left(T_1\right)U^{-1}\dots U{\mathsf{\Theta }}_n\left(T_n\right)U^{-1}.$$

Using unitarity and associativity,

$$=U{\mathsf{\Theta }}_n\left(T_1\right){\mathsf{\Theta }}_n\left(T_2\right)\dots {\mathsf{\Theta }}_n\left(T_n\right)U^{-1}={\mathrm{Ad}}_U\left({\mathsf{\Theta }}_n(T_1\dots T_n)\right).$$

So,

$${\Phi }_n(T_1\dots T_n)={\mathrm{Ad}}_U\left({\mathsf{\Theta }}_n(T_1\dots T_n)\right),$$

confirming that ${\Phi }_n={\mathrm{Ad}}_U\circ {\mathsf{\Theta }}_n$. Clearly, ${\mathsf{\Theta }}_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$, since it is a composition of n-automorphisms.

Moreover, since ${\Phi }_n$ and ${\mathrm{Ad}}_U$ preserve scalar multiples of the identity, ${\mathsf{\Theta }}_n$ must also satisfy ${\mathsf{\Theta }}_n\left(\lambda I\right)=\lambda I$. Thus, it acts trivially on $Z\left(\mathcal{A}\right)$.

Uniqueness follows from the uniqueness of spatial implementations of automorphisms in the strongly closed case. □

Theorem 14

(Structure of n-Automorphisms on Finite-Dimensional Nest Algebras). Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)\subseteq M_k\left(\mathbb{C}\right)$ be a finite-dimensional nest algebra associated with a finite chain of subspaces in ${\mathbb{C}}^k$. Then, every n-automorphism ${\Phi }_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$ satisfies the following:

(a) 

${\Phi }_n$ is implemented by conjugation with an invertible upper-triangular matrix in  $G{L}_k\left(\mathbb{C}\right)$;

(b) 

${\mathrm{Aut}}_n\left(\mathcal{A}\right)$ is isomorphic to a solvable subgroup of $G{L}_k\left(\mathbb{C}\right)$;

(c) 

Any two n-automorphisms are conjugate via an inner automorphism of $\mathcal{A}$.

Proof. 

(a)

Since $\mathcal{A}\subseteq M_k\left(\mathbb{C}\right)$ and consists of upper-triangular block matrices (due to the chain of subspaces forming $\mathcal{N}$), any automorphism ${\Phi }_n$ must preserve the triangular structure. In finite-dimensional settings, all automorphisms of nest algebras are inner and implemented via conjugation by an invertible element in $\mathcal{A}$ (see [2], Theorem 4.4.7).

Hence, for some $U\in G{L}_k\left(\mathbb{C}\right)$ with $U\mathcal{A}{U}^{-1}=\mathcal{A}$, we have

$${\Phi }_n\left(T\right)=UT{U}^{-1}\mathrm{for}\mathrm{all}T\in \mathcal{A}.$$

Because ${\Phi }_n$ preserves n-fold multiplicativity, this representation holds for all n-products as well.

(b)

Since the set of such conjugating matrices U lies within the normalizer of $\mathcal{A}$ in $G{L}_k\left(\mathbb{C}\right)$, and since $\mathcal{A}$ is solvable (as a triangular algebra), the group ${\mathrm{Aut}}_n\left(\mathcal{A}\right)$ inherits this solvable structure. Thus, ${\mathrm{Aut}}_n\left(\mathcal{A}\right)$ is isomorphic to a solvable subgroup of $G{L}_k\left(\mathbb{C}\right)$.

(c)

Let ${\Phi }_n,{\mathsf{\Psi }}_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$. Then there exist invertible matrices $U,V\in G{L}_k\left(\mathbb{C}\right)$ such that

$${\Phi }_n\left(T\right)=UT{U}^{-1},{\mathsf{\Psi }}_n\left(T\right)=VT{V}^{-1}.$$

Define $W=V^{-1}U\in G{L}_k\left(\mathbb{C}\right)$. Then,

$${\Phi }_n\left(T\right)=VWT{W}^{-1}{V}^{-1}={\mathrm{Ad}}_V\circ {\mathrm{Ad}}_W\left(T\right)={\mathsf{\Psi }}_n\circ {\mathrm{Ad}}_W\left(T\right).$$

Thus, ${\Phi }_n={\mathsf{\Psi }}_n\circ {\mathrm{Ad}}_W$, showing ${\Phi }_n$ and ${\mathsf{\Psi }}_n$ are conjugate via an inner automorphism.

□

4. Corollaries and Examples

In this section, we provide corollaries that follow naturally from the main theorems and illustrate the theoretical framework through concrete examples of n-derivations and n-automorphisms in finite- and infinite-dimensional nest algebras.

Corollary 1.

Let $\mathcal{A}=\mathrm{Alg}\left(\mathcal{N}\right)$ be a nest algebra on a separable infinite-dimensional Hilbert space. If $D_n:\mathcal{A}\to \mathcal{A}$ is a norm-continuous n-derivation such that $D_n\left(I\right)=0$ and $D_n\left(T^{*}\right)=D_n{\left(T\right)}^{*}$ for all $T\in \mathcal{A}$, then $D_n$ preserves every self-adjoint element of $\mathcal{A}$.

Corollary 2.

Let $\mathcal{A}\subseteq M_k\left(\mathbb{C}\right)$ be a finite-dimensional nest algebra and let ${\Phi }_n\in {\mathrm{Aut}}_n\left(\mathcal{A}\right)$. Then there exists an invertible upper-triangular matrix $U\in G{L}_k\left(\mathbb{C}\right)$ such that ${\Phi }_n\left(T\right)=UT{U}^{-1}$ for all $T\in \mathcal{A}$.

Example 1

(Finite-Dimensional Nest Algebra). Let $\mathcal{A}\subseteq M_3\left(\mathbb{C}\right)$ be the set of all upper-triangular matrices:

$$\mathcal{A}=\left\{\left(\begin{array}{ccc}a& b& c\\ 0& d& e\\ 0& 0& f\end{array}\right):a,b,c,d,e,f\in \mathbb{C}\right\}.$$

Define $D_3:\mathcal{A}\to \mathcal{A}$ by

$$D_3\left(T\right)=[A,T]+TAT-ATT,$$

where $A=\mathrm{diag}(1,2,3)$. Then $D_3$ is a three-derivation on $\mathcal{A}$. One can verify that $D_3$ preserves the triangular structure and maps nilpotent matrices to nilpotent matrices.

Example 2

(Infinite-Dimensional Nest Algebra). Let $\mathcal{H}={\ell }^2\left(\mathbb{N}\right)$ and let $\mathcal{N}$ be the standard nest consisting of closed spans of $\{e_1,\dots ,e_n\}$, where $\left\{e_n\right\}$ is the canonical basis of ${\ell }^2\left(\mathbb{N}\right)$. Then, $\mathrm{Alg}\left(\mathcal{N}\right)$ consists of infinite upper-triangular bounded operators.

Define ${\Phi }_n\left(T\right)=e^{AT}{e}^{-A}$ where A is a bounded diagonal operator such that $[A,T]$ remains in $\mathrm{Alg}\left(\mathcal{N}\right)$. Then, ${\Phi }_n$ is an n-automorphism, and the fixed-point subalgebra $\mathrm{Fix}\left({\Phi }_n\right)$ contains scalar multiples of the identity and diagonal matrices.

Example 3

(Automorphism from Jordan Block Conjugation). Let $J=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$ be a Jordan block matrix, and define ${\Phi }_2\left(T\right)=JT{J}^{-1}$ for $T\in M_2\left(\mathbb{C}\right)$.

Then ${\Phi }_2$ is a two-automorphism on the two-dimensional nest algebra of upper-triangular matrices. It preserves triangularity, and the associated fixed-point subalgebra consists of scalar diagonal matrices.

Example 4

(Non-Trivial n-Derivation on Operator Algebra). Let $\mathcal{H}={\ell }^2\left(\mathbb{N}\right)$, and define $D_n:B\left(\mathcal{H}\right)\to B\left(\mathcal{H}\right)$ by

$$D_n\left(T\right)=\sum _{j=1}^n{(-1)}^{j+1}{T}^{j}A{T}^{n-j},$$

for some compact diagonal operator A such that $D_n\left(T\right)\in \mathrm{Alg}\left(\mathcal{N}\right)$ for all T.

Then $D_n$ defines a bounded n-derivation on $\mathrm{Alg}\left(\mathcal{N}\right)$ which maps compact operators into compact operators and vanishes on identity.

Applications

The theory developed in this manuscript regarding n-derivations and n-automorphisms of nest algebras has several theoretical and potential applied implications. Below, we highlight key application domains where our results naturally extend or contribute to existing structures.

(a)

Triangular Operator Theory: Nest algebras play a central role in the study of triangular operators. Our results on triangularity preservation under n-derivations and exponential n-automorphisms contribute to the classification of operator models and refinement of similarity relations among triangular systems.

(b)

Invariant Subspace Problem: The invariant subspace problem, one of the central themes in operator theory, benefits from our fixed-point and ideal-preserving analysis. The structure of fixed-point subalgebras under n-automorphisms contributes to identifying operator substructures that remain invariant under higher-order symmetries.

(c)

Lie and Jordan Structures: As n-derivations generalize Lie algebraic derivations, our results bridge the theory of Lie and Jordan systems with functional analytic settings. The decomposition and commutator results extend naturally to generalized Jordan derivations on operator algebras.

(d)

Perturbation Theory and Stability Analysis: In perturbation theory, stability under symmetry transformations is critical. The preservation of radical, center, and ideals under n-maps contributes to understanding the robust behavior of algebraic systems under analytic or algebraic perturbations.

(e)

Quantum Computation and Non-Commutative Geometry: The structure of nest algebras arises in modeling quantum observables with partial commutativity. n-automorphism groups serve as symmetry operations, and the results here help identify classes of quantum symmetries preserving information subspaces.

(f)

Algorithmic Implications in Symbolic Computation: The explicit structural identities derived in this paper, such as exponential correspondence and fixed-point characterization, can be implemented in computer algebra systems to study nested operator algebras symbolically.

5. Conclusions

In this paper, we have introduced and systematically investigated the theory of n-derivations and n-automorphisms on nest algebras, a structure central to operator theory. Building upon foundational ideas from Lie algebras and derivations, our study extended these concepts through exponential mappings and higher-order transformation identities in both finite- and infinite-dimensional contexts.

We established multiple characterizations of structural invariance under n-maps, including the preservation of triangularity, radical and center stability, commutator identities, and the existence of fixed-point subalgebras. The group structure of n-automorphisms and the exponential correspondence from n-derivations to automorphisms were also presented with theoretical rigor and supported by examples.

Our findings not only enrich the abstract theory of higher-order maps in operator algebras but also bridge concepts from algebra, functional analysis, and non-commutative geometry. This work paves the way for further investigations into higher-order transformations on more general operator structures and their applications in mathematical physics and quantum computing.

Future Work and Limitations: One potential direction for future research involves extending the notion of n-derivations to $C^{*}$-algebras and von Neumann algebras where the topological structure may impose additional constraints. Further investigation into how n-maps interact with spectral properties, particularly essential spectra and joint spectra, could yield deeper insight into both operator theory and non-commutative geometry.