Fredholm theory of Toeplitz operators on standard weighted Fock spaces

We study the Fredholm properties of Toeplitz operators with bounded symbols of vanishing mean oscillation in the complex plane. In particular, we prove that the Toeplitz operator with such a symbol is Fredholm on a standard weighted Fock space if and only if the Berezin transform of the symbol is bounded away from zero outside a sufficiently large disk in the complex plane. We also show that the Fredholm index of the Toeplitz operator can be computed via the winding of the symbol along a sufficiently large circle. We finish by considering Toeplitz operators with matrix-valued symbols

Boundedness of Toeplitz Operators in Bergman-Type Spaces

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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

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