Extended Cesàro operators on Bergman spaces

We define an extended Cesàro operator Tg with holomorphic symbol g in the unit ball B of Cn. For a large class of weights w we characterize those g for which Tg is bounded (or compact) from Bergman space Lpa,w(B) to Lqa,w(B), 0<p,q<∞. In addition, we obtain some results about equivalent norms, the norm of point evaluation functionals, and the interpolation sequences on Lpa,w(B)

Localization and compactness of Operators on Fock Spaces

For $0<p\leq\infty$, let $F^{p}_\varphi$ be the Fock space induced by a weight function $\varphi$ satisfying $dd^c \varphi \simeq \omega_0$. In this paper, given $p\in (0, 1]$ we introduce the concept of weakly localized operators on $F^{p}_\varphi$, we characterize the compact operators in the algebra generated by weakly localized operators. As an application, for $0<p<\infty$ we prove that an operator $T$ in the algebra generated by bounded Toeplitz operators with $\textrm{BMO}$ symbols is compact on $F^p_\varphi$ if and only if its Berezin transform satisfies certain vanishing property at $\infty$. In the classical Fock space, we extend the Axler-Zheng condition on linear operators $T$, which ensures $T$ is compact on $F^p_{\alpha}$ for all possible $0<p<\infty$.Comment: 23 Page

Hankel operators on exponential Bergman spaces

We completely describe the boundedness and compactness of Hankel operators with general symbols acting on Bergman spaces with exponential type weights

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 with VMO symbols and the duality of generalized Fock spaces with small exponents

We characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents

Bounded, compact and Schatten class Hankel operators on Fock-type spaces

In this paper, we consider Hankel operators, with locally integrable symbols, densely defined on a family of Fock-type spaces whose weights are $C^3$-logarithmic growth functions with mild smoothness conditions. It is shown that a Hankel operator is bounded on such a Fock space if and only if its symbol function has bounded distance to analytic functions BDA which is initiated by Luecking(J. Funct. Anal. 110:247-271, 1992). We also characterize the compactness and Schatten class membership of Hankel operators. Besides, we give characterizations of the Schatten class membership of Toeplitz operators with positive measure symbols for the small exponent $0<p<1$. Our proofs depend strongly on the technique of H\"{o}mander's $L^2$ estimates for the $\overline{\partial}$ operator and the decomposition theory of BDA spaces as well as integral estimates involving the reproducing kernel

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

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