Toeplitz operators on Arveson and Dirichlet spaces

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Alpay D., Kaptanoglu H. T.

INTEGRAL EQUATIONS AND OPERATOR THEORY, vol.58, no.1, pp.1-33, 2007 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 58 Issue: 1
  • Publication Date: 2007
  • Doi Number: 10.1007/s00020-007-1493-1
  • Journal Name: INTEGRAL EQUATIONS AND OPERATOR THEORY
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1-33
  • Keywords: Toeplitz operator, weighted shift, m-isometry, unitary equivalence, Carleson measure, Berezin transform, Bergman metric, Bergman projection, weak convergence, Schatten-von Neumann ideal, Besov, Bergman, Dirichlet, Hardy, Arveson space, BESOV-SPACES, BERGMAN PROJECTIONS, COMPACT-OPERATORS, INTERPOLATION, THEOREM, HANKEL
  • Middle East Technical University Affiliated: No

Abstract

We define Toeplitz operators on all Dirichlet spaces on the unit ball of C-N and develop their basic properties. We characterize bounded, compact, and Schatten-class Toeplitz operators with positive symbols in terms of Carleson measures and Berezin transforms. Our results naturally extend those known for weighted Bergman spaces, a special case applies to the Arveson space, and we recover the classical Hardy-space Toeplitz operators in a limiting case; thus we unify the theory of Toeplitz operators on all these spaces. We apply our operators to a characterization of bounded, compact, and Schatten-class weighted composition operators on weighted Bergman spaces of the ball. We lastly investigate some connections between Toeplitz and shift operators.