A note on the Fredholm properties of Toeplitz operators on weighted Bergman spaces with matrix-valued symbols

Perälä, A. and Virtanen, J. A. (2011) A note on the Fredholm properties of Toeplitz operators on weighted Bergman spaces with matrix-valued symbols. Operators and Matrices, 5 (1). pp. 97-106. ISSN 1846-3886 doi: 10.7153/oam-05-06

Abstract/Summary

We characterize the essential spectra of Toeplitz operators Ta on weighted Bergman spaces with matrix-valued symbols; in particular we deal with two classes of symbols, the Douglas algebra C+H∞ and the Zhu class Q := L∞ ∩VMO∂ . In addition, for symbols in C+H∞ , we derive a formula for the index of Ta in terms of its symbol a in the scalar-valued case, while in the matrix-valued case we indicate that the standard reduction to the scalar-valued case fails to work analogously to the Hardy space case. Mathematics subject classification (2010): 47B35,