Extremal Domains for Self-Commutators in the Bergman Space

Abstract

In [10], the authors have shown that Putnam's inequality for the norm of self-commutators can be improved by a factor of 1 2 for Toeplitz operators with analytic symbol ϕ acting on the Bergman space A 2 (Ω). This improved upper bound is sharp when ϕ(Ω) is a disk. In this paper we show that disks are the only domains for which the upper bound is attained.

Hyponormal Toeplitz operators with non-harmonic Symbol acting on the Bergman space

“…and it is a standing conjecture that the univalent condition can be dropped. Evidence for this conjecture was given in [7] where it was showed that…”

“…[12, p. 263]) that the norm of the commutator of T * ϕ and T ϕ is bounded above by Area(ϕ(D))/π for analytic ϕ, and in [11] it was shown that this bound can be improved to Area(ϕ(D))/(2π) for analytic and univalent ϕ. In [7], it was conjectured that the hypothesis "univalent" is superfluous for this stronger bound. We extend this conjecture to non-analytic symbols.…”

Selected problems in classical function theory

“…This is why the statement is restricted to simply connected domains. The authors of [12] noted that refining Olsen and Reguera's proof implies that the equality for the selfcommutator upper bound in simply connected domains holds only for disks.…”

“…One might then ask what happens in spaces other than E 2 . The authors in [12] explore this question for Bergman spaces, following the paper [3]. Without loss of generality, we can assume there exists a measure µ in C such that T is unitarily equivalent to the operator T z of multiplication by z on L 2 a (µ), the closure (in L 2 (µ)) of functions analytic in a neighborhood of the support of µ (see [3]).…”

A note on the spectral area of Toeplitz operators

“…Recently, there has been revived interest in the topic in the context of analytic Topelitz operators on the Bergman space (cf. [3], [10] and [7]). Together with Putnam's inequality, the latter lower bounds have provided an alternative proof of the celebrated St. Venant's inequality for torsional rigidity.…”