Spectral theory of Toeplitz and Hankel operators on the Bergman space A1

The Fredholm properties of Toeplitz operators on the Bergman space A2 have been well-known for continuous symbols since the 1970s. We investigate the case p=1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on Ap that arise when we no longer have 1<p<∞; in particular bounded Toeplitz operators on A1 were characterized completely very recently but only for bounded symbols. We also consider compactness of Hankel operators on A1

Boundedness of Toeplitz Operators in Bergman-Type Spaces

Peer reviewe

Weak BMO and Toeplitz operators on Bergman spaces

Inspired by our previous work on the boundedness of Toeplitz operators, we introduce weak BMO and VMO type conditions, denoted by BWMO and VWMO, respectively, for functions on the open unit disc of the complex plane. We show that the average function of a function f is an element of BWMO is boundedly oscillating, and the analogous result holds for f is an element of VWMO. The result is applied for generalizations of known results on the essential spectra and norms of Toeplitz operators. Finally, we provide examples of functions satisfying the VWMO condition which are not in the classical VMO or even in BMO.Peer reviewe

Boundedness of Toeplitz Operators in Bergman-Type Spaces

Peer reviewe

Weak BMO and Toeplitz operators on Bergman spaces

Inspired by our previous work on the boundedness of Toeplitz operators, we introduce weak BMO and VMO type conditions, denoted by BWMO and VWMO, respectively, for functions on the open unit disc of the complex plane. We show that the average function of a function f is an element of BWMO is boundedly oscillating, and the analogous result holds for f is an element of VWMO. The result is applied for generalizations of known results on the essential spectra and norms of Toeplitz operators. Finally, we provide examples of functions satisfying the VWMO condition which are not in the classical VMO or even in BMO.Peer reviewe

On the Berger-Coburn phenomenon

In their previous work, the authors proved the Berger-Coburn phenomenon for compact and Schatten $S_p$ class Hankel operators $H_f$ on generalized Fock spaces when $1<p<\infty$, that is, for a bounded symbol $f$, if $H_f$ is a compact or Schatten class operator, then so is $H_{\bar f}$. More recently J.~Xia has provided a simple example that shows that there is no Berger-Coburn phenomenon for trace class Hankel operators on the classical Fock space $F^2$. Using Xia's example, we show that there is no Berger-Coburn phenomena for Schatten $S_p$ class Hankel operators on generalized Fock spaces $F^2_\varphi$ for any $0<p\le 1$. Our approach is based on the characterization of Schatten class Hankel operators while Xia's approach is elementary and heavily uses the explicit basis vectors of $F^2$, which cannot be found for the weighted Fock spaces that we consider. We also formulate four open problems

Fredholm Toeplitz Operators on Doubling Fock Spaces

Recently the authors characterized the Fredholmn properties of Toeplitz operators on weighted Fock spaces when the Laplacian of the weight function is bounded below and above. In the present work the authors extend their characterization to doubling Fock spaces with a subharmonic weight whose Laplacian is a doubling measure. The geometry induced by the Bergman metric for doubling Fock spaces is much more complicated than that of the Euclidean metric used in all the previous cases to study Fredholmness, which leads to considerably more involved calculations.Peer reviewe

Schatten class Hankel operators on the Segal-Bargmann space and the Berger-Coburn phenomenon

We characterize Schatten $p$-class Hankel operators $H_f$ on the Segal-Bargmann space when $0<p<\infty$ in terms of our recently introduced notion of integral distance to analytic functions in $\mathbb{C}^n$. Our work completes the study inspired by a theorem of Berger and Coburn on compactness of Hankel operators and subsequently initiated twenty years ago by Xia and Zheng, who obtained a characterization of the simultaneous membership of $H_f$ and $H_{\overline f}$ in Schatten classes $S_p$ when $1\le p<\infty$ in terms of the standard deviation of $f$. As an application, we give a positive answer to their question of whether $H_f\in S_p$ implies $H_{\overline{f}}\in S_p$ when $f\in L^\infty$ and $1<p<\infty$, which was previously solved for $p=2$ and $n=1$ by Xia and Zheng and for $p=2$ in any dimension by Bauer in 2004. In addition, we prove our results in the context of weighted Segal-Bargmann spaces, which include the standard and Fock-Sobolev weights