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Hyponormal Toeplitz operators with polynomial symbols on weighted Bergman spaces
Journal of Inequalities and Applications volume 2014, Article number: 335 (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.
Let be the open unit disk in the complex plane. For , the weighted Bergman space of the unit disk is the space of analytic functions in , where
The space is a Hilbert space with the inner product
If then is the Bergman space . For any nonnegative integer n, let
where stands for the usual Gamma function. It is easy to check that is an orthonormal basis for . For , the Toeplitz operator , and the Hankel operator on are defined by
where denotes the orthogonal projection that maps from onto . The reproducing kernel in is given by
for . We thus have
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 .
For , write
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 ()
Let , where . If and , then
In this note we consider the hyponormality of Toeplitz operators on with symbol in the class of functions with polynomials f and g. Since the hyponormality of operators is translation invariant we may assume that . The following relations can easily be proved:
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.
Lemma 1 () For any s, t nonnegative integers,
For , write
The following two lemmas will be used for proving the main result of this section.
Lemma 2 For , we have
Proof Let . Then we have
This proves (i). For (ii), if then by Lemma 1 we have
If instead , a similar argument gives the result. □
Lemma 3 ()
Let and . If , then for , we have
Our main result now follows.
Theorem 4 Let , where
If and , then on is hyponormal if and only if
Proof For , put
Then a straightforward calculation shows that is hyponormal if and only if
Also we have
Substituting (2.2) and (2.3) into (2.1), it follows from Lemma 3 that
Therefore it follows from Lemma 2 that is hyponormal if and only if
Then a direct calculation gives
Therefore (2.4) and (2.5) show that is hyponormal if and only if
This completes the proof. □
Remark 5 Let , where
If , , and on is hyponormal. Then
Proof If we let for and the other ’s be 0 into (2.4), then we have
Then is a strictly decreasing function and
Therefore (2.6) and (2.7) give that if is hyponormal then
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.
Remark 7 Let , where and are nonzero. Suppose on is hyponormal. It is well known  that
However, the result cannot be extended to the case of ; for example, if then a straightforward calculation shows that the selfcommutator of Toeplitz operator on is given by
where . Thus and the trace of the selfcommutator .
Zhu K: Theory of Bergman Spaces. Springer, New York; 2000.
Cowen C: Hyponormality of Toeplitz operators. Proc. Am. Math. Soc. 1988, 103: 809-812. 10.1090/S0002-9939-1988-0947663-4
Curto RE, Hwang IS, Lee WY: Hyponormality and subnormality of block Toeplitz operators. Adv. Math. 2012, 230: 2094-2151. 10.1016/j.aim.2012.04.019
Curto RE, Lee WY: Joint hyponormality of Toeplitz pairs. Mem. Am. Math. Soc. 2001. 150: Article ID 712
Hwang IS, Lee WY: Hyponormality of trigonometric Toeplitz operators. Trans. Am. Math. Soc. 2002, 354: 2461-2474. 10.1090/S0002-9947-02-02970-7
Hwang IS, Kim IH, Lee WY: Hyponormality of Toeplitz operators with polynomial symbol. Math. Ann. 1999, 313: 247-261. 10.1007/s002080050260
Lu Y, Shi Y: Hyponormal Toeplitz operators on the weighted Bergman space. Integral Equ. Oper. Theory 2009, 65: 115-129. 10.1007/s00020-009-1712-z
Lu Y, Liu C: Commutativity and hyponormality of Toeplitz operators on the weighted Bergman space. J. Korean Math. Soc. 2009, 46: 621-642. 10.4134/JKMS.2009.46.3.621
Nakazi T, Takahashi K: Hyponormal Toeplitz operators and extremal problems of Hardy spaces. Trans. Am. Math. Soc. 1993, 338: 759-769.
Sarason D: Generalized interpolation in . Trans. Am. Math. Soc. 1967, 127: 179-203.
Hwang IS, Lee JR: Hyponormal Toeplitz operators on the weighted Bergman spaces. Math. Inequal. Appl. 2012, 15: 323-330.
Lee J, Lee Y: Hyponormality of Toeplitz operators on the weighted Bergman spaces. Honam Math. J. 2013, 35: 311-317. 10.5831/HMJ.2013.35.2.311
Farenick DR, Lee WY: Hyponormality and spectra of Toeplitz operators. Trans. Am. Math. Soc. 1996, 348: 4153-4174. 10.1090/S0002-9947-96-01683-2
Hwang IS: Hyponormal Toeplitz operators on the Bergman spaces. J. Korean Math. Soc. 2005, 42: 387-403.
Farenick DR, Lee WY: On hyponormal Toeplitz operators with polynomial and circulant-type symbols. Integral Equ. Oper. Theory 1997, 29: 202-210. 10.1007/BF01191430
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.
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Hwang, I.S., Lee, J. & Park, S.W. Hyponormal Toeplitz operators with polynomial symbols on weighted Bergman spaces. J Inequal Appl 2014, 335 (2014). https://doi.org/10.1186/1029-242X-2014-335
- Toeplitz operators
- weighted Bergman space
- polynomial symbols