Finite Rank Perturbations of Toeplitz Products on the Bergman Space

DOI10.1016/J.JFA.2020.108850arXiv2011.05414MaRDI QIDQ6353467

Trieu Minh Nhut Le, Damith Thilakarathna

Publication date: 10 November 2020

Abstract: In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution

d i a m o n d

on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if

F_{j}, G_{j}

(

1 l e q j l e q N

) are polynomials of

z

and

then

s u m_{j = 1}^{N} T_{F_{j}} T_{G_{j}} - T_{H}

is a finite rank operator for some

L^{1}

-function

H

if and only if

s u m_{j = 1}^{N} F_{j} d i a m o n d G_{j}

belongs to

L^{1}

and

H = s u m_{j = 1}^{N} F_{j} d i a m o n d G_{j}

. In the case

F_{j}

's are holomorphic and

G_{j}

's are conjugate holomorphic, it is shown that

H

is a solution to a system of first order partial differential equations with a constraint.

Mathematics Subject Classification ID

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