Toeplitz Operators on Harmonic Fock Spaces with Radial Symbols

: The main aim of this paper is to study new features and specific properties of the Toeplitz operator with radial symbols in harmonic Fock spaces. A new spectral decomposition of a Toeplitz operator with Wick symbols is also established


Introduction and Preliminaries
Let C denote the complex plane and dµ = dv (z)  π e z z be the Gaussian measure on C, where dv(z) = dx dy is the standard Lebesgue plane measure on C ≃ R 2 .Denote by L 2 (C, dµ) the Hilbert space of square Lebesgue integrable functions on C with dµ.The Fock space F 2 (C) is the Hilbert space of analytic functions which belong to L 2 (C, dµ).In fact, the Fock space F 2 (C) is a reproducing function space of the reproducing kernel The harmonic Fock space F 2 h (C) is the subspace of L 2 (C, dµ) consisting of all harmonic functions on C. As in the harmonic Bergman space, it is known (see, e.g., [1]) that where The space F 2 h (C) is also a reproducing function space with the reproducing kernel R z (w) = K z (w) + K z (w) − 1, z, w ∈ C.
The concept and properties of Fock space have been improved and generalized in many various different directions by several authors; for more details, see, e.g., [1][2][3][4] and the references therein.
Let Q be the harmonic Bargmann projection from L 2 (C, dµ) into harmonic Fock space F 2 h (C).For a function a ∈ L 2 (C, dµ), the Toeplitz operator T a : F 2 h (C) → F 2 h (C) with symbol a is the linear operator defined by where T a is densely defined and not bounded in general.
The theory of the Toeplitz operator stems from a series of ideas developed in the second half of the 20th century by mathematicians such as Otto Toeplitz, Hermann Hankel, Eberhard Hopf, Norbert Wiener and Gábor Szegö.In recent years, many scholars have obtained many important results for the Toeplitz operator T a and the Hankel operator in Fock space F 2 (C).The boundedness, compactness, algebraic properties and the product properties on Fock-type spaces have been studied extensively.For more details, we refer the readers to, for example, [1,3,[5][6][7][8][9][10][11][12][13][14][15].Because the product of two harmonic functions is often no longer a harmonic function, it is more difficult to study Toeplitz operators in harmonic Fock space than in analytic Fock space.Many of the methods used for operators in the analytic Fock space lose their effectiveness in harmonic Fock space.Therefore, many scholars attempted to provide some new ideas and methods to overcome such situations and generalize Fock spaces.For example, in the paper [2], Chen et alia considered the Toeplitz operator T a in vector-valued generalized Fock spaces.In the paper [4], He and Wu characterized dual Toeplitz operators in the orthogonal complement of Fock-Sobolev spaces.
There have been some results of the Toeplitz operator T a in harmonic Bergman space; please refer to [16][17][18].In the paper [16], Guo and Zheng characterized compact Toeplitz operators in the unit disk D. In [18], by using the system of integral equations, Lee characterized the commuting Toeplitz operators of holomorphic symbols and pluriharmonic symbols in pluriharmonic Bergman space.In addition, Lee proved in [17] the commutativity of an operator with a radial symbol and an operator with a pluriharmonic symbol in pluriharmonic Bergman space.In the papers [7, [19][20][21][22], several mathematicians analyzed the influence of the radial component of a symbol of the spectral, compactness and Fredholm properties of Toeplitz operators on Bergman space or Fock space.In the paper [21], Li and Lu characterized radial symbols in Bergman spaces over the polydisk.In [22], the first author of this paper and her coauthor gave some properties about Toeplitz operators in weighted pluriharmonic Bergman spaces with radial symbols.
In this paper, inspired by the above, we are committed to investigating the problem of Toeplitz operators with radial symbols in a harmonic Fock space F 2 h (C).Basing our work on the techniques used in [7, 19,20,22], we will construct an operator R whose restriction in harmonic Fock space , where I is the identity operator.Employing the operator R, we will prove that each Toeplitz operator T a with radial symbols is unitary to the multiplication operator γ a I. Making use of the Berezin concept of Wick and anti-Wick symbols (see [1,5,23]), we will show that, in our particular case (radial symbols), the Wick symbols of a Toeplitz operator give complete information about the operator and provide its spectral decomposition.

New Results for Harmonic Fock Spaces and Related Operators
We now recall some notations, definitions and well-known facts about harmonic Fock space.Let f (z) ∈ F 2 h (C) and write f = g + h.The harmonic Fock space F 2 h (C) can be described as the closure in L 2 (C, dµ) of the set of all smooth functions satisfying the equations where z = x + y i.As performed in [7], introduce the unitary operator with the rule Then the image dx dy), which satisfies the equations Passing to polar coordinates in R 2 , we have where S 1 is the unit circle and is the element of length.In addition, ∂ ∂r As performed in [7], introduce the unitary operator and the discrete Fourier transform F : The inverse of F is given by It is not difficult to see that ∂ ∂r , where Z + = {0} ∪ N and Z − = Z \ Z + .Thus, the image F (2) = U 2 (F (1) ) of the space F (1)  can be described as the subspace of L 2 (R + , r dr) ⊗ l 2 , which is the closure of all sequences {c n (r)} n∈Z with smooth components satisfying the equations 1 2 and 1 2 Equations ( 1) and ( 2) are easy to solve and their general solutions have the form (3) and Each function c n (r) must be in L 2 (R + , r dr).Therefore, the space F (2) coincides with the space of all two-sided sequences {c n (r)} n∈Z with For each n ∈ Z, as performed in [7], introduce the unitary operator where As performed in [7], define the unitary operator The space F (3) = U 3 (F (2) ) coincides with the space of all sequences {d n (r)} n∈Z , where Let l 0 (r) = 1 e r/2 .Then ℓ 0 (r) ∈ L 2 (R + ) and ∥ℓ 0 (r)∥ = 1.Denote by L 0 the onedimensional subspace of L 2 (R + ) generated by l 0 (r).Then the one-dimensional projection P 0 of L 2 (R + ) onto L 0 is of the form Now, F (3) = L 0 ⊗ ℓ 2 and the orthogonal projection 3) is obviously of the form P (3) = P 0 ⊗ I.The above work leads to the following theorem: Theorem 1.The unitary operator U = U 3 U 2 U 1 gives an isometric isomorphism of the space L 2 (C, dµ) onto L 2 (R + ) ⊗ ℓ 2 such that the following statements hold: (a) The harmonic Fock space , where L 0 is the one-dimensional subspace of L 2 (R + ) generated by the function ℓ 0 (r) = 1 e r/2 .(b) The harmonic Bargmann projection is the unitary equivalent of U Q U −1 = P 0 ⊗ I, where P 0 is the one-dimensional projection of L 2 (R + ) onto L 0 .
Proof.First, we prove (a).For any f ∈ F 2 h (C) and f = g + h, according to the definitions of U 1 , U 2 and U 3 , as well as Equations ( 3)-(7), we have which shows (a).
To see (b), according to the definitions of Q and U, we obtain So, it follows from Equation (7) that which shows (b).The proof of Theorem 1 is completed.
Introduce the isometric imbedding The adjoint operator R * 0 : is an isometric isomorphism.The adjoint operator is an isometric isomorphism of l 2 in the subspace Proof.Let {c n } ∈ ℓ 2 .Then we have The proof of Theorem 2 is thus complete.

Toeplitz Operators with Radial Symbols
Denote by L ∞ 1 R + , 1 e r 2 the linear space of all measurable functions a(r) on R + for which the integrals In this section, we investigate Toeplitz operators with symbols from acting in the harmonic Fock space F 2 h (C).
Then the Toeplitz operator T a acting in harmonic Fock space F 2 h (C) is the unitary equivalent of the multiplication operator γ a I acting on ℓ 2 .The sequence γ a = {γ a (n)} n∈Z is given by Proof.The operator T a is obviously the unitary equivalent of the operator where the function α n (r) is given by (5) and Theorem 3 is proved.
Theorem 4. The Toeplitz operator T a with radial symbol a = a(r) belonging to L Proof.This follows directly from Theorem 3. Proposition 1 below implies that a Toeplitz operator T a with symbol is a well-defined linear operator with a dense domain.
Proposition 1.Let F 2 0 (C) and F 2 0 (C) denote the set of all polynomials on z and z, respectively.If c n r n t n e (rz)/t dt t a(r)r e r 2 dr Accordingly, we acquire and the set F 2 0 (C) + F 2 0 (C) is the domain for each Toeplitz operator T a with symbol a(r).The proof of Proposition 1 is thus complete.
By Proposition 1, the Toeplitz operator T a with symbol a(r) ∈ L ∞ 1 R + , 1 e r 2 has a bounded extension to the whole space F 2 h (C) if and only if the sequence {γ a (n)} is bounded.
Corollary 3. The spectrum of a bounded Toeplitz operator T a is given by sp T a = {γ a (n) : n ∈ Z} and its essential spectrum ess sp T a coincides with the set of all limit points of the sequence {γ a (n)} n∈Z .

Properties of Toeplitz Operators with Radial Symbols
We start with conditions which guarantee the boundedness or compactness of Toeplitz operators with radial symbols from L ∞ 1 R + , 1 e r 2 .
Theorem 5. Let a(r) ∈ L ∞ 1 R + , 1 e r 2 .Then the Toeplitz operator T a is bounded on F 2 h (C) if one of the following statements holds: (3) The function Proof.The first statement is well known.Let the second statement hold.Then we have a (n).
Accordingly, the second statement is proved.Finally, integrating by parts yields Further applying the second statement to the function B(r) leads to the third statement.Theorem 5 is thus proved.
Theorem 6.Let a(r) ∈ L ∞ 1 R + , 1 e r 2 .Then the Toeplitz operator T a is compact on F 2 h (C) if one of the following statements holds: (1) The limit lim r→∞ a(r) = 0 is valid.
(2) The limits Proof.This follows directly from Theorem 5.
A Toeplitz operator T a with a symbol a = a(z) acting in the harmonic Fock space F 2 h (C) is an operator with the anti-Wick symbol a = a(z).The function ã(z, z) is called a Wick symbol of an operator T if this operator acts on F 2 h (C) as follows: The Wick and anti-Wick symbols of the same operator are connected by the formula Denote by M the linear subspace of L ∞ 1 R + , 1 e r 2 such that for each a(r) ∈ M the Toeplitz operator T a(r) is bounded on F 2 h (C).Denote by T (M) the C * -algebra generated by all Toeplitz operators T a with symbols a ∈ M.
The system of functions is an orthonormal basis for the harmonic Fock space F 2 h (C).Denote by L (n) the onedimensional space generated by the function l (n) (z).The orthogonal projection P (n) : They are Toeplitz operators with the symbol from M.
For any n ∈ Z, the one-dimensional space L (n) is an eigenspace for the Toeplitz operator T a with a(r) ∈ M, and the corresponding eigenvalue is equal to γ a (n).The required proof is complete.

Conclusions
This paper is devoted to studying specific properties (such as the boundedness, compactness, algebraic properties, spectral decomposition and others) of the Toeplitz operator with radial symbols in harmonic Fock spaces.On the basis of analytic functions theory, we present several problems of harmonic functions and expand the scope of the past study.In summary, new important results and features for Toeplitz operators with radial symbols in harmonic Fock spaces are established (see Theorems 5-7).We believe that these newly discovered results will help us study the problems in pluriharmonic Fock spaces or polydisk Fock spaces in future studies.
harmonic Fock space F 2 h (C) if and only if e r−u du = 0 are valid.

Theorem 7 .
Let a(r) ∈ M. Writing the Toeplitz operator T a in the form of an operator with a Wick symbol gives the spectral decomposition of the operator T a :γ a (n)P (n) .Proof.For n ∈ Z and f ∈ F 2 h (C), consider the operator with the Wick symbol of the form p (n) (z, z) = e z z z n zn n! .Then (T a f )(γ a (n)(P (n) f )(z).