Hyponormal Toeplitz operators with polynomial symbols on weighted Bergman spaces
© Hwang et al.; licensee Springer. 2014
Received: 16 May 2014
Accepted: 26 August 2014
Published: 2 September 2014
In this note we consider the hyponormality of Toeplitz operators on weighted Bergman space with symbol in the class of functions with polynomials f and g.
for and .
A bounded linear operator A on a Hilbert space is said to be hyponormal if its selfcommutator is positive (semidefinite). The hyponormality of Toeplitz operators on the Hardy space of the unit circle has been studied by Cowen , Curto, Hwang and Lee [3–5] and others . Recently, in  and , the hyponormality of on the weighted Bergman space was studied. In , Cowen characterized the hyponormality of Toeplitz operator on by properties of the symbol . Here we shall employ an equivalent variant of Cowen’s theorem that was first proposed by Nakazi and Takahashi .
Then is hyponormal if and only if is nonempty.
The solution is based on a dilation theorem of Sarason . For the weighted Bergman space, no dilation theorem (similar to Sarason’s theorem) is available. In , the first named author characterized the hyponormality of on in terms of the coefficients of the trigonometric polynomial φ under certain assumptions as regards the coefficients of φ on the weighted Bergman space when and in , extended for all .
Theorem A ()
The purpose of this paper is to prove Theorem A for the Toeplitz operators on when f and g of degree N.
2 Main result
In this section we establish a necessary and sufficient condition for the hyponormality of the Toeplitz operator on the weighted Bergman space under a certain additional assumption concerning the symbol φ. The assumption is related on the symmetry, so it is reasonable in view point of the Hardy space . We expect that this approach would provide some clue for the future study of the symmetry case.
The following two lemmas will be used for proving the main result of this section.
If instead , a similar argument gives the result. □
Lemma 3 ()
Our main result now follows.
This completes the proof. □
This completes the proof. □
Example 6 Let and . Then by Theorem A, is not hyponormal. But φ satisfies the inequality in Remark 5, hence the inverse of Remark 5 is not satisfied.
where . Thus and the trace of the selfcommutator .
This work was supported by National Research Foundation of Korea Grant funded by the Korean Government (2011-0022577). The authors are grateful to the referee for several helpful suggestions.
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