Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 769-783

FREDHOLM THEORY OF TOEPLITZ OPERATORS ON STANDARD WEIGHTED FOCK SPACES

Aamena Al-Qabani and Jani A. Virtanen

University of Reading, Department of Mathematics
Whiteknights, Reading RG6 6AX, England; rn030601 'at' reading.ac.uk

University of Reading, Department of Mathematics
Whiteknights, Reading RG6 6AX, England; j.a.virtanen 'at' reading.ac.uk

Abstract. We study the Fredholm properties of Toeplitz operators with bounded symbols of vanishing mean oscillation in the complex plane. In particular, we prove that the Toeplitz operator with such a symbol is Fredholm on a standard weighted Fock space if and only if the Berezin transform of the symbol is bounded away from zero outside a sufficiently large disk in the complex plane. We also show that the Fredholm index of the Toeplitz operator can be computed via the winding of the symbol along a sufficiently large circle. We finish by considering Toeplitz operators with matrix-valued symbols.

2010 Mathematics Subject Classification: Primary 47B35; Secondary 30H20.

Key words: Toeplitz operators, Hankel operators, Fock spaces, Fredholm properties, essential spectrum, compactness, block Toeplitz operators, matrix-valued symbols.

Reference to this article: A. Al-Qabani and J. A. Virtanen: Fredholm theory of Toeplitz operators on standard weighted Fock spaces. Ann. Acad. Sci. Fenn. Math. 43 (2018), 769-783.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4344