The Trace Of The Curvature And The N-hypercontractivity Of Operators

The classification of operators is an important topic in operator theory.The clas-sification of operators includes unitary equivalence and similarity equivalence.It is very difficult to judge the unitary equivalence(or similarity equivalence)of any two bound-ed linear operators,so one can only consider the relatively special subclasses of opera-tors.In 1978,M.J.Cowen and R.G.Douglas defined a class of geometric operators:Cowen-Douglas operators,and proved that the curvature and its covariant derivatives are unitary invariants,but the relation between the curvature and similarity equivalence of Cowen-Douglas operators is unclear.Later,many experts studied similarity equivalence of Cowen-Douglas operators and gave some beautiful results.For example,R.G.Dou-glas,H.Kwon and S.Treil used the model theorem to describe the similarity equivalence of n-hypercontractive operators and the direct sum of weighted backward shift opera-tors.However,up to now,the geometric similarity invariants of general Cowen-Douglas operators have not been solved effectively.At the same time,determining when an operator is subnormal is always one of the difficult problems in operator theory.When J.Agler generalized the Arveson extension theorem,he introduced the notion of the n-hypercontraction and proved that a bounded operator T is subnormal contraction if and only if for any positive integer n,T is n-hypercontraction.And we know that every bounded linear operator can be multiplied by an appropriate constant to become a contraction.Therefore,it is very necessary to study the n-hypercontractivity of operators,and this study is quite difficult.We also know that the n-hypercontractivity of operators has an important application in the classification of similarity equivalence of operators.In view of this,we need to explore the sufficient and necessary conditions for an operator to be an n-hypercontraction,and construct corresponding examples to illustrate that n-hypercontractivity of operators in the conclusion of similarity equivalence of operators by R.G.Douglas,H.Kwon and S.Treil is a necessary condition.In view of the above questions,this paper is divided into the following three parts:In the first part,we study the relationship between the n-hypercontractivity of the weighted backward shift operators and its weight sequences,and give the necessary and sufficient conditions for the weighted backward shift operator to be n-hypercontraction by using the weight sequence.In the second part,we characterize the n-hypercontractivity of operators by using the trace of the curvature and give the concrete application of the n-hypercontraction and the trace of the curvature in the similarity equivalence.In addition,we construct corresponding examples to illustrate the relationship between the n-hypercontractivity of operators and the similarity equivalence,and prove that the n-hypercontractivity of operators in the conclusion of similarity equivalence of operators by R.G.Douglas,H.Kwon and S.Treil is a necessary condition.In the third part,we give a sufficient condition for similarity of any Cowen-Douglas operators with index one.