Special orthonormal basis for $L^2$ functions on the unit circle

Young-Bok Chung

doi:10.4134/BKMS.b160697

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2017; 54(6): 2013-2027

Online first article July 7, 2017 Printed November 30, 2017

https://doi.org/10.4134/BKMS.b160697

Special orthonormal basis for $L^2$ functions on the unit circle

Young-Bok Chung

Chonnam National University

Abstract

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Abstract

We compute explicitly the matrices represented by Toeplitz operators on the Hardy space over the unit circle with respect to a special orthonormal basis constructed by author in terms of their symbols. And we also find a necessary condition for the matrix generated by the product of two Toeplitz operators with respect to the basis to be a Toeplitz matrix by a direct calculation and we finally solve commuting problems of two Toeplitz operators in terms of symbols. This is a generalization of the classical results obtained regarding to the orthonormal basis consisting of the monomials.

Keywords: Toeplitz operator, Toeplitz matrices, Hardy space, Szego kernel

MSC numbers: Primary 47B35, 30C40