Generalized Hankel operators and the generalized solution operator to on the Fock space and on the Bergman space of the unit disc

✍ Scribed by Wolfgang Knirsch; Georg Schneider

• Publisher: John Wiley and Sons
• Year: 2006
• Tongue: English
• Weight: 168 KB
• Volume: 279
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200410446

✦ Synopsis

Abstract

In this paper we consider Hankel operators $ \tilde H _{{\bar z}^k}$ = (Id – P ~1~)$ \bar z^k $ from A ^2^(ℂ, |z |^2^) to A ^2,1^(ℂ, |z |^2^)^⊥^. Here A ^2^(ℂ, |z |^2^) denotes the Fock space

A ^2^(ℂ, |z |^2^) = {f: f is entire and ‖f ‖^2^ = ∫~ℂ~ |f (z)|^2^ exp (–|z |^2^) dλ (z) < ∞}.

Furthermore A ^2,1^(ℂ, |z |^2^) denotes the closure of the linear span of the monomials {$ \bar z ^l $z ^n^ : n, l ∈ ℕ, l ≤ 1} and the corresponding orthogonal projection is denoted by P ~1~. Note that we call these operators generalized Hankel operators because the projection P ~1~ is not the usual Bergman projection. In the introduction we give a motivation for replacing the Bergman projection by P ~1~. The paper analyzes boundedness and compactness of the mentioned operators.

On the Fock space we show that $ \tilde H _{{\bar z}^2}$ is bounded, but not compact, and for k ≥ 3 that $ \tilde H _{{\bar z}^k}$ is not bounded. Afterwards we also consider the same situation on the Bergman space of the unit disc. Here a completely different situation appears: we have compactness for all k ≥ 1.

Finally we will also consider an analogous situation in the case of several complex variables. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)