Weighted Shift Operators And Conjugate Symmetry

As a concrete and important class of operators,weighted shift operators acting on Hilbert space play an important role in the research of operator theory and provide examples for study-ing operator structure and properties.Many new theories in operator theory are verified their correctness by the weighted shift operators.In these last years,the study of the properties of weighted shift operators acting on Hilbert space is a hot topic.It has received extensive attention from operator theorists at home and abroad.The theorists continue to enrich the properties of weighted shift operators.This paper primarily provides the definition of conjugate symmetric operators and studies approximation problems of conjugate symmetry.Let H be a complex separable infinite dimensional Hilbert space.An operator T acting on H is said to be conjugate symmetric,if there exists an operator A such that T is unitarily equiv-alent to A⊕A~*.In this paper,by using the weighted sequences,we completely characterize the conjugate symmetry of unilateral weighted shifts.We prove that the conjugate symmetry is not stable under small compact perturbations for any bounded operator.And at the same time,we characterize the conjugate symmetry of normal operators and which normal operators can become conjugate symmetry under small compact perturbations.Furthermore,we characterize which normal operator have arbitrarily small compact perturbation to be conjugate symmetric.We prove that the class of the conjugate symmetric operators is not norm closed.The content of this article and its structure will be further described below.This article is divided into two parts.The first section mainly elaborates on the research context and the states of research for weighted shifts.We present preliminary knowledge,common symbols in this paper.In addi-tion,we introduce the background about the research on weighted shifts.We briefly present important results about the research on weighted shifts by scholars at home and abroad in recent years.The second section is the main section of this paper.In this part,we give the definition of conjugate symmetry and the main conclusions.Additionally,we characterize the conjugate symmetry of all unilateral weighted shifts and part of bilateral weighted shifts by using the weighted sequences.We prove that the conjugate symmetry is not stable under small compact perturbations for any bounded operator.Later in this chapter,we give the characterization of the conjugate symmetry of normal operators.we characterize which normal operator have arbitrarily small compact perturbation to be conjugate symmetric.We prove that the class of the conjugate symmetric operators is not norm closed.In this part,we also give the proof of these theorems and some conclusions.In addition,we provide several examples.The third section is a summary of the main results of this paper.