Averaged wave operators and complex-symmetric operators

TL;DR

This paper investigates the convergence properties of sequences involving unitary operators with singular spectral measures, exploring their connection to complex-symmetric operators and providing conditions for convergence in a specific model case.

Contribution

It introduces a conjecture on the weak averaged convergence of certain operator differences and links this to the theory of complex-symmetric operators, offering new insights and conditions for convergence.

Findings

Conjecture on weak averaged convergence of operator sequences.

Connection established between convergence and complex-symmetric operators.

Sufficient conditions for convergence in a model case involving integral kernels.

Abstract

We study the behaviour of sequences $U_2^n X U_1^{-n}$, where $U_1, U_2$ are unitary operators, whose spectral measures are singular with respect to the Lebesgue measure, and the commutator $XU_1-U_2X$ is small in a sense. The conjecture about the weak averaged convergence of the difference $U_2^n X U_1^{-n}-U_2^{-n} X U_1^n$ to the zero operator is discussed and its connection with complex-symmetric operators is established in a general situation. For a model case where $U_1=U_2$ is the unitary operator of multiplication by $z$ on $L^2(\mu)$, sufficient conditions for the convergence as in the Conjecture are given in terms of kernels of integral operators.