Complex Analysis and Operator Theory

, Volume 10, Issue 6, pp 1213–1226 | Cite as

Averaged Wave Operators and Complex-symmetric Operators

Article

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.

Keywords

Wave operators Singular spectral measure Cesàro means 

Mathematics Subject Classification

Primary 47B38 Secondary 47A58 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Chebyshev Laboratory, Department of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  2. 2.St. Petersburg Department of the Steklov Mathematical InstituteSt. PetersburgRussia

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