Remarks on Hyponormal Toeplitz Operators on the Weighted Bergman Spaces
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Complex Analysis and Operator Theory
Remarks on Hyponormal Toeplitz Operators on the Weighted Bergman Spaces Eungil Ko1 · Jongrak Lee2 Received: 8 May 2020 / Accepted: 10 October 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper, we study the hyponormal Toeplitz operators Tϕ , where ϕ is a trigonometric polynomial symbol with finite degree. We present necessary and sufficient conditions for the hyponormality of Tϕ under some assumptions about the coefficients of ϕ. Next, we find the necessary condition for hyponormality of Tϕ . Keywords Toeplitz operators · Hyponormal · Weighted Bergman space Mathematics Subject Classification 47B20 · 47B35
1 Introduction Let D be the open unit disk in the complex plane C and let d A be the normalized area measure on D. For −1 < α < ∞, we define the weighted Bergman space A2α (D) as the space of analytic functions in L 2 (D, d Aα ), where d Aα (z) = (α + 1)(1 − |z|2 )α d A(z). Here, the space L 2 (D, d Aα ) is a Hilbert space with the inner product f , gα =
D
f (z)g(z)d Aα (z)
( f , g ∈ L 2 (D, d Aα )).
Communicated by Scott McCullough.
B
Jongrak Lee [email protected] Eungil Ko [email protected]
1
Department of Mathematics, Ewha Womans University, Seoul 03760, Korea
2
Department of Mathematics, Jeju National University, Jeju 63243, Korea 0123456789().: V,-vol
83
Page 2 of 19
E. Ko , J. Lee
For any n ∈ N ∪ {0} and z ∈ D, let en (z) =
(n + α + 2) zn , (n + 1)(α + 2)
where (s) is the usual gamma function. It is well-known that {en } is an orthonormal set in A2α (D) ([7]). Since the set of polynomials is dense in A2α (D), we can easily check that {en } is an orthonormal basis for A2α (D). If f (z) =
∞
an z
n
and g(z) =
n=0
∞
bn z n
n=0
are two functions in A2α (D), then f , g =
∞ (n + 1)(2 + α) n=0
(n + α + 2)
an b n .
Let P denote the orthogonal projection from L 2 (D, d Aα ) onto A2α (D). For ϕ ∈ L ∞ (D), the Toeplitz operator Tϕ with symbol ϕ on A2α (D) is defined by Tϕ f :=P(ϕ · f ) where f is in A2α (D) and the Hankel operator Hϕ with symbol ϕ on A2α (D) is defined by Hϕ f = (I − P)(ϕ · f ). A bounded linear operator T on a Hilbert space is said to be hyponormal if its selfcommutator [T ∗ , T ]:=T ∗ T − T T ∗ is positive (semidefinite). Many authors have been extensively researched about hyponormality of Toeplitz operators. On the Hardy space H 2 (T) of the unit circle T = ∂D, the hyponormality of Toeplitz operators has been studied by many authors in [1,3,4,6,8,17]. In [1], using the properties of the symbol ϕ ∈ L ∞ (T), Cowen characterized the hyponormality of Tϕ on H 2 (T) as follows. Cowen’s Theorem. [1] For ϕ ∈ L ∞ (T), let E(ϕ) := {k ∈ H ∞ : ||k||∞ ≤ 1 and ϕ − kϕ ∈ H ∞ (T)}. Then Tϕ is hyponormal if and only if E(ϕ) is nonempty. The main idea of proof is a dilation theorem of Sarason [18]. But, the above theorem can not be applied to the weighted Bergman space. Recently, many results for the hyponormality of Tϕ on A2α (D) have been obtained; see [2,5,9,13–16]. In particular, in [9,16], the hyponormality o
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