Contractivity and expansivity of H-Toeplitz operators on the Bergman spaces

: In this paper we consider the properties of H-Toeplitz operators B ϕ on the Bergman space L 2 a ( D ). We present some necessary and su ﬃ cient conditions for the contractive and expansive H-Toeplitz operators B ϕ with various symbols ϕ


Introduction
Let D denote the open unit disk in the complex plane C and dA the area measure on the complex plane C. The space L 2 (D) is a Hilbert space with the inner product The Bergman space L 2 a (D) consists of all analytic functions on D and L ∞ (D) is the space of the essentially bounded measurable function on D. For ϕ ∈ L ∞ (D), the multiplication operator M ϕ on L 2 a (D) is defined by M ϕ ( f ) = ϕ • f and the Toeplitz operator T ϕ on L 2 a (D) is defined by where P denotes the orthogonal projection of L 2 (D) onto L 2 a (D) and f ∈ L 2 a (D).It is clear that those operators are bounded if ϕ ∈ L ∞ (D).
The harmonic Bergman space L 2 harm (D) denotes the space of all complex-valued harmonic functions in L 2 (D).The space L 2 harm (D) is a closed subspace of L 2 (D) and it is a Hilbert space.Let P harm be the orthogonal projection from the space L 2 (D) onto the space L 2 harm (D).
Toeplitz operators on the Bergman space were studied by McDonald and Sundberg in [19].Recently, lots of research about Toeplitz operators has been conducted in the Bergman space (see [2,11]).In the Hardy space, the hyponormality of Toeplitz operators was studied in [7,8,12,14,20] ; refer to references therein for more details.Recently, many authors characterized the hyponormality of Toeplitz operators on the Bergman space and weighted Bergman space (see [7,13,15,16,18,21]).In 2007, Arora and Paliwal [1] have introduced the notion of H-Toeplitz operators on the Hardy space.Recently, in [10], the authors studied H-Toeplitz operators on the Bergman space.The research of H-Toeplitz operators has arisen naturally in several fields of mathematics and in a variety problems.For example, an H-Toeplitz system comprises a matrix equation of the form T x = y where T is an n by n H-Toeplitz matrix with x, y in C n .The n × n H-Toeplitz matrix T has 2n − 1 degrees of freedom rather than n 2 .Thus for a large n, it is easier to solve the system of linear equations for an H-Toeplitz matrix(cf.[10]).In this paper we consider the algebraic properties of H-Toeplitz operators B ϕ on the Bergman space L 2 a (D).More concretely, we establish a tractable and explicit criterion for the contractivity and expansivity of H-Toeplitz operators.Several decades ago, many researchers began studying the contractive and expansive operators (see [3,4,5,6]).In [5], the authors considered the invariant subspace problem for contractive operators.Recently, various results have been derived based on the papers (see [9,17]).
The organization of this paper is as follows.In Section 2, we introduce the notion of H-Toeplitz operators on the Bergman space and provide various well-known properties of these operators.In Section 3, we focus on the contractive and expansive H-Toeplitz operators with analytic, coanalytic and harmonic symbols.

Preliminaries and auxiliary lemmas
Let s, t be nonnegative integers and P be the orthogonal projection from L 2 (D) to L 2 a (D).Then we have The following lemmas will be used frequently in this paper.
Lemma 2.1.([10]) In the harmonic Bergman space L 2 harm (D), for nonnegative integers s and t, the following: Lemma 2.2.( [15]) For m ≥ 0, we have By using Lemmas 2.1 and 2.2, we have the following result.
Remark 2.3.For m ≥ 0, we have In order to define the notion of an H-Toeplitz operator on L 2 a (D), we first consider the operator for all n ≥ 0 and z ∈ D. It can be checked that the operator K is bounded linear on L 2 a (D) with ||K|| = 1.Moreover, the adjoint K * of the operator K is given by K * (e n (z)) = e 2n (z) and K * (e n+1 (z)) = e 2n+1 (z) for all n ≥ 0. From the definition of K and K * , we have that KK * = I L 2 harm (D) and Remark 2.4.By the definitions of K and K * , we can easily check that K(z 2n ) = Next, we define H-Toeplitz operators on the Bergman space L 2 a (D) using the definition of the operator K.
But in the case of the H-Toeplitz operator, Therefore, B * z (az) B z (az).A straightforward calculation shows that B z B z B z 2 (cf.[10]).

Main results
A bounded linear operator T on a Hilbert space is said to be expansive if T * T ≥ I, contractive if T * T ≤ I, and isometric if T * T = I. where

H-Toeplitz operators with analytic symbols
In this subsection, we consider the properties of H-Toeplitz operators B ϕ and B * ϕ with analytic symbols.First, we study the contractivity and expansivity of B ϕ and B * ϕ with ϕ = az N for N ∈ N and a ∈ C. Next, we extend the symbol ϕ of the form ϕ(z) = ∞ i=0 a i z i with a i ∈ C. Theorem 3.1.Let ϕ(z) = az N for N ∈ N and a ∈ C. Then B ϕ is contractive if and only if |a| ≤ 1.
and we have that According to the definition for the contractivity of B ϕ , the inequality There are two cases to consider.If c 0 for is even and c = 0 for is odd, from (3.1), we have or equivalently, n+1 is decreasing for n, we have If c 0 for is odd, and c = 0 for is even, from (3.1), we have or equivalently, Set c 2N+1 0 and c i = 0 for i 2N + 1.Then from (3.5), 1 2(N+1) ≤ 0; it is a contradiction.In the next result, we have the sufficient condition for the contractivity and expansivity of the H-Toeplitz operators B ϕ with symbols ϕ for any nonnegative integer s.
Proof.For any k ∈ L 2 a (D), for any c j ∈ C ( j = 0, 1, 2, • • • ).Then on comparing the coefficient of z m , by the equation (3.7) we have that Set c 0 for some and c j = 0 for any j .Then we consider that the following two cases arise: Case 1: If = 2s for any nonnegative integer s, then for any nonnegative integer s.Case 2: If = 2s + 1 for any nonnegative integer s, then Thus, for any nonnegative integer s.This completes the proof.
and so, B ϕ is not contractive.
The following example shows that the converse of Theorem 3.3 (ii) is not true.
From Theorem 3.6, we get the following corollaries and example.Proof.From the proof of Theorem 3.6, B * ϕ is expansive if and only if Hence, the inequality given by (3.9) holds for any arbitrary 4 .By a direct calculation, Since c i 's are arbitrary, set c 0 0 and c i = 0 for i > 0; then Proof.In the proof of Theorems 3.1 and 3.6, Then, on comparing the coefficient of z 0 , we get Since c 2N−1 and c N are arbitrary, B ϕ is not self-adjoint.
Corollary 3.10.Let ϕ(z) = az N for N ∈ N and a ∈ C. Then B ϕ is not normal. Proof.
As in the proof of Theorems 3.1 and 3.6, we have Since c i 's are arbitrary, set c 2N+1 0 and In the next result, we investigated a sufficient condition for the contractivity and expansivity of the adjoint H-Toeplitz operators B * ϕ with symbols ϕ(z) = ∞ i=0 a i z i where for any nonnegative integer s.
Proof.For any k ∈ L 2 a (D), Set c s 0 for some s and c j = 0 for any j s.Then Thus, This completes the proof.
The following example shows that the converse of Theorem 3.11 (ii) is not true.
Example 3.12.Consider the polynomial ϕ(z) Then the condition given by (3.10) Proof.We have the result by putting s = 0 in Theorem 3.11.
and by Corollary 3.13, B * ϕ is not contractive.

H-Toeplitz operators with coanalytic symbols
In this subsection, we consider the properties of H-Toeplitz operators B ϕ and B * ϕ with coanalytic, or antianalytic symbols.First, we study the contractivity and expansivity of B ϕ and B * ϕ with ϕ = bz N for N ∈ N and b ∈ C. Next, we extend the symbol ϕ of the form ϕ Hence B ϕ is contractive if and only if If we compare the coefficients of c 2n+2N , we have for any n ≥ 0; thus, This completes the proof.Proof.From the proof of Theorem 3.15, B ϕ is expansive if and only if Since c i 's (0 ≤ i < 2N) are arbitrary, we put c i 0 if i is odd and c i = 0 if i is even; then, 0 ≥ 1 i+1 ; it is a contradiction.
In the next result, we get a sufficient condition for the contractivity of H-Toeplitz operators B ϕ with symbols ϕ for any s ∈ N.
Proof.For any k ∈ L 2 a (D), Then on comparing the coefficient of z m , by the equation (3.11), we have that We set c 0 if = 2s and c = 0 if 2s for some s ∈ N. Thus B ϕ on L 2 a (D) is contractive then Therefore, s i=1 (s This completes the proof.
On the other hand, we have that The following theorem is purposed to find the necessary and sufficient conditions for the contractivity of the adjoint H-Toeplitz operator B * ϕ with coanalytic symbols ϕ.Proof.For any k ∈ L 2 a (D), Thus, B * ϕ on L 2 a (D) is contractive if and only if This completes the proof.
From Theorem 3.19, we get the following corollary and example.Proof.From the proof of Theorem 3.19, B * ϕ is expansive if and only if or equivalently, ϕ is expansive if and only if Since c i 's are arbitrary, we set c 0 0 and c i = 0 for i > 0; then, ||B * ϕ k(z)|| In view of Corollaries 3.9 and 3.10, we have the following result.In the next theorem, we have the necessary and sufficient condition for the contractivity and expansivity of adjoint H-Toeplitz operators B * ϕ with symbols ϕ Proof.For any k ∈ L 2 a (D), Then B * ϕ is contractive if and only if Similarly, B * ϕ is expansive if and only if This completes the proof.
for any nonnegative integer s.

H-Toeplitz operators with harmonic symbols
Finally, we study the properties of H-Toeplitz operators B ϕ with harmonic symbols of the form ϕ(z) = ∞ i=0 a i z i + ∞ i=1 b i z i with a i , b i ∈ C. Specifically, we focus on the necessary and sufficient conditions of contractivity and expansivity for B ϕ and B * ϕ , respectively.Theorem 3.25.Let ϕ for any nonnegative integer s.
Proof.By a similar argument as in the proof of Theorems 3.3 and 3.17, for any k ∈ L 2 a (D), for any c j ∈ C ( j = 0, 1, 2, • • • ).Set c 0 for some and c j = 0 for any j .Then we consider the following two cases: Case 1: If = 2s for any nonnegative integer s and c 2s 0 then for any nonnegative integer s.
(ii) If B * ϕ is expansive, then for any nonnegative integer s.
Proof.By a similar argument as in the proof of Theorems 3.11 and 3.23, for any k ∈ L 2 a (D), for any c j ∈ C ( j = 0, 1, 2, • • • ).Set c s 0 for some s and c j = 0 for any j s.Then for any s ∈ N.

Conclusions
We characterized the necessary or sufficient conditions for the contractive and expansive H-Toeplitz operators B ϕ with various symbols ϕ on the Bergman space L 2 a (D).By these results, we expect to provide the properties of these operators on the Bergman space.

Definition 2 . 5 .Remark 2 . 7 .
([10]) For ϕ ∈ L ∞ (D), the H-Toeplitz operator B ϕ with the symbol ϕ is defined as the operator B ϕ :L 2 a (D) → L 2 a (D) such that B ϕ ( f ) = PM ϕ K( f ) for all f ∈ L 2 a(D).The next proposition follows from the definition of the H-Toeplitz operators.Proposition 2.6.([10]) For ϕ, ψ ∈ L ∞ (D), the operator B ϕ satisfies the following: (i) B ϕ is a bounded linear operator on L 2 a (D) with ||B ϕ || ≤ ||ϕ|| ∞ .(ii) For any scalar α and β, B αϕ+βψ = αB ϕ + βB ψ .(iii) The adjoint of the H-Toeplitz operator B ϕ is given by B * ϕ = K * P harm M ϕ .The following remark provides important information for adjoint operators.It shows the difference between adjoint Toeplitz operators and adjoint H-Toeplitz operators.If f, g are in L ∞ (D) then by the definition of Toeplitz operators T f , we have that

Corollary 3 . 7 .
Let ϕ(z) = az N for N ∈ N and a ∈ C. Then B * ϕ is expansive if and only if |a| 2 ≥ N + 1.

Corollary 3 . 16 .
Let ϕ(z) = bz N for N ∈ N and b ∈ C. Then B ϕ is neither expansive nor isometric.

Theorem 3 . 19 .
Let ϕ(z) = bz N for N ∈ N and b ∈ C. Then B * ϕ is contractive if and only if |b| ≤ 1.

Corollary 3 .
22. Let ϕ(z) = bz N for N ∈ N and b ∈ C. Then B * ϕ is neither self-adjoint nor normal.