Research Article | Open Access
Zhi Ling Sun, Yu Feng Lu, "Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols", Abstract and Applied Analysis, vol. 2011, Article ID 210596, 15 pages, 2011. https://doi.org/10.1155/2011/210596
Toeplitz Operators on the Weighted Pluriharmonic Bergman Space with Radial Symbols
We construct an operator whose restriction onto weighted pluriharmonic Bergman Space is an isometric isomorphism between and . Furthermore, using the operator we prove that each Toeplitz operator with radial symbols is unitary to the multication operator . Meanwhile, the Wick function of a Toeplitz operator with radial symbol gives complete information about the operator, providing its spectral decomposition.
Let be the open unit ball in the complex vector space . For any and in , let , where is the complex conjugate of and . For a multi-index and , we write , where , and is its length, .
The weighted pluriharmonic Bergman space is the subspace of the weighted space consisting of all pluriharmonic functions on . A pluriharmonic function in the unit ball is the sum of a holomorphic function and the conjugate of a holomorphic functions. It is known that is a closed subspace of and hence is a Hilbert space. Let be the Hilbert space orthogonal projection from onto . For a function , the Toeplitz operator with symbol is the linear operator defined by
is densely defined and not bounded in general.
The boundedness and compactness of Toeplitz operators on Bergman type spaces have been studied intensively in recent years. The fact that the product of two harmonic functions is no longer harmonic adds some mystery in the study of Toeplitz operators on harmonic Bergman space. Many methods which work for the operator on analytic Bergman spaces lost their effectiveness on harmonic Bergman space. Therefore new ideas and methods are needed. We refer to [1–3] for references about the results of Toeplitz operator on harmonic Bergman space. The paper  characterizes compact Toeplitz operators in the case of the unit disk . In , the authors consider Toeplitz operators acting on the pluriharmonic Bergman space and study the problem of when the commutator or semicommutator of certain Toeplitz operators is zero. Lee  proved that two Toplitz operators acting on the pluriharmonic Bergman space with radial symbols and pluriharmonic symbol, respectively, commute only in an obvious case.
The authors in  analyze the influence of the radial component of a symbol to spectral, compactness and Fredholm properties of Toeplitz operators on Bergman space on unit disk . In , they are devoted to study Toeplitz operators with radial symbols on the weighted Bergman spaces on the unit ball in .
In this paper, we will be concerned with the question of Toeplitz operators with radial symbols on the weighted pluriharmonic Bergman space. Based on the techniques in [4–6], we construct an operator whose restriction onto weighted pluriharmonic Bergman space is an isometric isomorphism between and , and where is the subspace of . Using the operator we prove that each Toeplitz operator with radial symbols is unitary to the multication operator acting on . Next, we use the Berezin concept of Wick and anti-Wick symbols. It turns out that in our particular (radial symbols) case the Wick symbols of a Toeplitz operator give complete information about the operator, providing its spectral decomposition.
2. Pluriharmonic Bergman Space and Orthogonal Projection
We start this section with a decomposition of the space . Consider a nonnegative measurable function , such that , and where is the surface area of unit sphere and is the Gamma function.
Introduce the weighted space where is the usual Lebesgue volume measure and is the space with the usual Lebesgue surface measure.
The space is the direct sum of mutually orthogonal spaces , that is, where denotes the space of spherical harmonics of order . Meanwhile, each space is the direct sum (under the identification ) of the mutually orthogonal spaces (see, e.g., ): where , for each , is the space of harmonic polynomials (their restrictions to the unit sphere) of complete order in the variable and complete order in the conjugate variable . Thus, we can get
The Hardy space in the unit ball is a closed subspace of . Denote by the Szegö orthogonal projection of onto the Hardy space . It is well known that . The standard orthonormal base in has the form
Fix an orthonormal basis , , in the space so that , , .
Passing to the spherical coordinates in the unit ball we have For any function have the decomposition with the coefficients satisfying the condition
by the rule , and
Let be a pluriharmonic in the unit ball and write , where the functions , are holomorphic in . Suppose
are their power series representations of and , respectively. We have where , , , .
Let be the pluriharmonic Bergman space in from . Denote by the pluriharmonic Bergman orthogonal projection of onto the Bergman space . From the above it follows that to characterize a function and considering its decomposition according to (2.13), one can restrict to the function having the representation Now let us take an arbitrary from in the form (2.14). It will satisfy the Cauchy-Riemann equations, that is,
Applying , to and , respectively, we have
where , and we come to the infinite system of ordinary linear differential equations
Their general solution has the form , , with . Hence, for any we have
And, it is easy to verify . Thus the image is characterized as the closed subspace of which consists of all sequences of the form
For each introduce the function Obviously, there exists the inverse function for the function on , which we will denote by . Introduce the operator
By Proposition 2.1 in , the operator maps unitary onto in such a way that
Intoduce the unitary operator where
By (2.23), we can get the space coincides with the space of all sequences for which
Let . We have and . Denote by the one-dimensional subspace of generated by . The orthogonal projection of onto has the form Let . Denote by the subspace of consisting of all sequences . And let be the orthogonal projections of onto , then , where , if and , if .
Observe that and the orthogonal projection of
onto has the form . This leads to the following theorem.
Theorem 2.1. The unitary operator gives an isometric isomorphism of the space onto such that
(1) the pluriharmonic Bergman space is mapped onto , where is the one-dimensional subspace of , generated by the function ;
(2) the pluriharmonic Bergman projection is unitary equivalent to where is the one-dimensional projection (2.27) of onto .
Introduce the operator
by the rule
The mapping is an isometric embedding, and the image of coincides with the space . The adjoint operator
is given by
Meanwhile the operator maps the space onto , and its restriction
is an isometric isomorphism. The adjoint operator
is isometric isomorphism of onto the subspace of .
Remark 2.2. We have
Theorem 2.3. The isometric isomorphism is given by
Proof. Let , we can get
Corollary 2.4. The inverse isomorphism is given by where , , , and is the standard basis for the pluriharmonic Bergman space ; that is,
3. Toeplitz Operator with Radial Symbols on
In this section we will study the Toeplitz operators with radial symbols .
Theorem 3.1. Let be a measurable function on the segment . Then the Toeplitz operator acting on is unitary equivalent to the multication operator acting on . The sequence is given by
(i) The Toeplitz operator with measurable radial symbol is bounded on if and only if . Moreover,
(ii) The Toeplitz operator is compact if and only if . The spectrum of the bounded operator is given by and its essential spectrum ess-sp coincides with the set of all limits points of the sequence .
Let be a Hilbert space and a subset of elements of parameterized by elements of some set with measure .
Then is called a system of coherent states, if for all ,
or equivalently, if for all ,
We define the isomorphic inclusion by the rule
By (3.7) we have , where and are the scalar products on and , respectively, and . Let . A function is an element of if and only if for all , . The operator is the orthogonal projection of onto .
The function , , is called the anti-Wick (or contravariant) symbol of an operator if
or, in other terminology, the operator is the Toeplitz operator with the symbols .
Given an operator , introduce the (Wick) function
If the operator has an anti-Wick symbols, that is, for some function , then
And the operator admits the following representation in terms of its Wick function: Interchanging the integrals above, we understand them in a weak sense.
The restriction of the function onto the diagonal
is called the Wick (or covariant or Berezin) symbols of the operator .
The Wick and anti-Wick symbols of an operator are connected by the Berezin transform
The pluriharmonic Bergman reproducing kernel in the space has the form where . For , the reproducing property
shows that the system of functions , , forms a system of coherent states in the space . In our context, we have , , , , , where .
Lemma 3.3. Let be the Toeplitz operator with a radial symbol . Then the corresponding Wick function (3.11) has the form
Denote by the one-dimensional subspace of generated by the base element , . Then the one-dimensional projection of onto has obviously the form In the similar method, denote the one-dimensional subspace of generated by the base element . Let be the projection from onto , and the projection can be rewritten as
Theorem 3.4. Let be a bounded Toeplitz operator having radial symbol . Then one can get the spectral decomposition of the operator :
The value depends only on . Collecting the terms with the same and using the formula
we obtain where . The orthogonal projection of onto the subspace generated by all element with , can be written as
denotes the orthogonal projection from onto the subspace generated by all elements with . Therefore, (3.22) has the form
In view of (3.25), we can get the following useful corollary.
Corollary 3.5. Let be a bounded Toeplitz operator having radial symbol . Then the Wick symbol of the operator is radial as well and is given by the formula where .
In terms of Wick function the composition formula for Toeplitz operators is quite transparent.
Corollary 3.6. Let , be the Toeplitz operators with the Wick function respectively. Then the Wick function of the composition is given by
This work was supported by the National Science Foundation of China (Grant no. 10971020).
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Copyright © 2011 Zhi Ling Sun and Yu Feng Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.