Multipliers on Hilbert spaces of functions on R

TL;DR

This paper generalizes classical multiplier representation theorems for Hilbert spaces of functions on R, providing new insights into the structure of multipliers and the spectrum of the shift operator.

Contribution

It extends the multiplier representation theorem to broader Hilbert spaces included in L^1_{loc}(R) and describes the spectrum of the shift operator in this context.

Findings

Representation theorem for multipliers commuting with shift operator

Description of the spectrum of the shift operator

Generalization of classical multiplier results

Abstract

For a Hilbert space H included in L^1_{loc} (R) of functions on $R we obtain a representation theorem for the multipliers M commuting with the shift operator S. This generalizes the classical result for multipliers in L^2(R) as well as our previous result for multipliers in weighted space Lw^2(R). Moreover, we obtain a description of the spectrum of S.