Averaged wave operators and complex-symmetric operators

Roman Bessonov; Vladimir Kapustin

doi:10.1007/s11785-015-0496-1

paper

Averaged wave operators and complex-symmetric operators

arXiv:1504.03820 · doi:10.1007/s11785-015-0496-1

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 $X U_{1} - U_{2} X$ 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 $L^{2} (μ)$ , sufficient conditions for the convergence as in the Conjecture are given in terms of kernels of integral operators.

arXiv comment13 pages