Roots of Toeplitz operators on the Bergman space

Issam Louhichi and Nagisetty V. Rao

Vol. 252 (2011), No. 1, 127–144

DOI: 10.2140/pjm.2011.252.127

Abstract

A major goal in the theory of Toeplitz operators on the Bergman space over the unit disk 𝔻 in the complex plane ℂ is to competely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. In [2007], the first author characterized the commutant of a Toeplitz operator T that has a quasihomogeneous symbol ϕ(r)e^ip𝜃 with p > 0, in case it has a Toeplitz p-th root S with symbol ψ(r)e^i𝜃: The commutant of T is the closure of the linear space generated by powers Sⁿ that are Toeplitz. But the existence of a p-th root was known until now only when ϕ(r) = r^m with m ≥ 0. Here we will show the existence of p-th roots for a much larger class of symbols, for example, those symbols for which

Keywords

Toeplitz operators, Bergman space, Mellin transform, Gamma funtion, Beta function

Mathematical Subject Classification 2000

Primary: 47B35

Secondary: 47L80

Milestones

Received: 3 July 2010

Revised: 18 December 2010

Accepted: 21 December 2010

Published: 8 October 2011

Proposed: Robert Finn