Toeplitz and Hankel operators on Bergman spaces

Abstract

In this paper we consider Toeplitz and Hankel operators on the Bergman spaces of the unit ball and the polydisk in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper C Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {C}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose symbols are bounded measurable functions. We give necessary and sufficient conditions on the symbols for these operators to be compact. We study the Fredholm theory of Toeplitz operators for which the corresponding Hankel operator is compact. For these Toeplitz operators the essential spectrum is computed and shown to be connected. We also consider symbols that extend to continuous functions on the maximal ideal space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript normal infinity Baseline left-parenthesis normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi mathvariant="normal">∞</mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^\infty }(\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; for these symbols we describe when the Toeplitz or Hankel operators are compact.