Similarity And Approximation Of Complex Symmetric Operators

In this dissertation,H will always denote a complex separable Hilbert space of infinite dimension.We let B(H)denote the algebra of all bounded linear operators on H.We let K(H)denote the ideal of compact operators in B(H).Definition 1([51]).A map C on is called an antiunitary operator if C is conjugate-linear,invertible and 〈Cx,Cy〉=〈y,x〉,(?)x,y ∈H-If in addition,C-1=C,then C is called a conjugation.Definition 2([51]).An operator T G B(H)is said to be complex symmetric if there exists a conjugation C on H such that CTC=T*Complex symmetry operators are widely used in many fields of basic mathematics and applied mathematics as well as quantum mechanics,study-ing complex symmetry operators has far-reaching significance.However,the current operator theory is still incomplete and further research is needed.In 1968,Halmos[37]proved that the set of irreducible operators is a dense subset set of B(H).After that,Halmos[38]raised a question whether every operator is the limit of a reducible operator or not.In order to answer this question,Voiculescu[9]proved the famous Weyl-von Neumann theo-rem,and gave a positive answer to Halmos question.Inspired by the results of Halmos-Voiculescu,now the first aim of the study is to prove the approximation of complex symmetric operators.Ac-cording to study of reducible operators and irreducible operators([9][37]),it is natural to ask:is every complex symmetric operator the limit of a reducible operator or an irreducible operator?In this dissertation,we will give a complete answer,that is the following theorem.Theorem 3.(a)If dim H<∞,then ICSO = CSO and RCSO is anowhere dense closed subset of CSO.(b)If dim H =∞,then RCSOI=ICSO=CSO.In the above theorem,CSO is the set of all complex symmetric operators on H.We denote by RCSO the set of all reducible complex symmetric oper-ators on H,and ICSO the set of all irreducible complex symmetric operators on H.When dim H=∞,we got the following result.Theorem 4.If T ∈B(H)is complex symmetric,then T is approximately unitarily equivalent to a reducible complex symmetric operator.Voiculescu proves that each operator on the complex divisible infinite dimensional Hilbert space is approximately equivalent to a reducible opera-tor.Therefore the above result is a complex symmetric version of the result of Voiculescu.Normal operators are complex operators,Gilbreath and Wogen studied the polar decomposition of complex symmetric operators in[69],and it is proved that the complex symmetry operator can be regarded as a general-ization of normal operators in a certain sense.In view of the important role of the normal operator theory in the study of operator algebra theory,nat-urally we care about the connection between complex symmetric operators and operator algebra research.In 2017,Zhu Sen and Shen Junhao[23]s-tudied which von Neuman algebras or C*-algebras can be generated by a single complex symmetry operator.Inspired by their results,we hope to s-tudy which operator T generated C*-algebra C*(T)can be generated by a single complex symmetry operator.For the case where T is an essential normal operator,this paper gives a complete answer to the above question.In particular,when T is an irre-ducible operator,we have the following result.Theorem 5.If T∈∈B is an irreducible essential normal operator,then C*(T)can be generated by a complex symmetric operator if and only if ind[T-z)=0,(?)z(?)σe Here σe(T)represents the essential spec-trum of T.At the end of this dissertation is to study the stability of complex sym-metry under similarity.It is can be seen from[51,Ex.4],each finite rank operator is similar to a complex symmetry operator.Therefore,there is a complex symmetry operator in the space where the dimension is greater than or equal to 2,the operator which is similar to it is not necessarily complex symmetric.Naturally,we have to ask:Which complex symmetry of the op-erator is constant under the similar transformation?In this paper,we have the following results,giving a complete answer to the above questionsTheorem 6.ForT ∈ B(H),the following are equivalent:(i)S(T)(?)CSO;(ii)S(T)(?)CSO;(iii)S(T)(?)CSO;(iv)T is an algebraic operator of degree at most two.In the above theorem,S(T)represents a similar orbit of T ∈ B(H),that is,the set of all operators similar to T on S(T).