Strongly Irreducible Operators

In this paper,C is the complex plane and H denotes a complex separable Hilbert space.B(H)denotes the algebras of all bounded linear operators acting on H.K(H)denotes the ideal of all compact operators on H and LatT denotes the lattice of invariant subspaces of T∈B(H).F.Gilfeather introduced the notion of strongly irreducibility of operator in 1972.An operator T is called a strongly irreducible operator if T is not similar to any reducible operator.In 1979,Professor Jiang Zejian proposed the notions of Banach reducible operator and Banach irreducible operator.An operator T is called a Banach reducible operator if there exist two nontrivial invariant subspaces M and N of T such that M∩N = {0} and M + N = H; otherwise,T is called Banach irreducible.It is an easy exercise to prove for an operator T that T is strongly irreducible if and only if T is Banach irreducible.The Jordan canonical form of a matrix plays an important role in matrix theory.And if dimH＜∞,then an operator T∈B(H)is Banach irreducible if and only if there is only one Jordan block in the Jordan canonical form of T.Since Banach irreducible operators,or briefly,BIR operators,and Jordan blocks on a finite dimensional space have many features in common,Jiang Zejian considers BIR operators as the approximate replacement of Jordan blocks on infinite dimensional Hilbert space and hopes that a theorem similar to the Jordan canonical form theorem in matrix theory can be set up with this replacement. Several problems concerning with BIR operators were studied at Jilin University and finally approximate Jordan canonical form theorem has been set up on separable infinite dimensional space and nest algebra.In the approximation associated with BIR operators,people always hope to use a special class of operators to approximate general operators in certain way.What Question H concern is that if BIR operator(of course spectrum -connected)could approximate operators with connected spectrum,furthermore, this approximation is "small compact" approximation,as to this question, people at first consider "small compact" approximation in a smaller class of operators,until 2002,Professor Ji Youqing complete answered the question H.Similar classification is a basic question in operator theory,as far as general Operator are concern,On the one hand,seeking the Jordan canonical form theorem on infinite dimensional spaces;On the other hand,Be certain degree of understanding Through appropriate weakening or strengthening the similarity. D.A.Herrero considered U + K similar which is between Unitary equivalence and similar,quasidiagonal,essentially normal operators are stable in U + K similar,One of the most typical achievements,made by C.Apostol,L.A.Fialkow, D.A.Herrero and D.Voiculescu is the theorem of similarity orbit of operators.This theorem suggests that the fine spectral picture is the complete similarity invariant as far as the closure of similarity orbit of operators are concerned.Essentiallysimilar and quasisimilar are weaker equivalent relation than similar,people studied Essentiallysimilar and quasisimilar of strongly irreducible operator,Professor Ji Youqing Characterized essentialsimilar invariant of strongly irreducible operators;Chunlan Jiang and Hua He got quasisimilar transformation preserve strongly irreducibly about Cowen-Douglas operators;Chunlan Jiang,Yongfei Jin and Zongyao Wang proved similar and quasisimilar coincide on operator weighted shifts and gave a necessary and sufficient conditions about similar of strongly irreducibly operator weighted shifts.But,with the research going deeper and deeper,people are badly in need of new ideas and new tools to be introduced so as to further their research. In 1980,G.Elloitt classified AF-algebra successfully by using of K-theory language,which stimulated people want to describe similarity invariant of operators in the term of the ordered K-group of their commutant algebras.In 1978,M.J.Cowen and R.G.Douglas defined a class of geometrical operators, Cowen-Douglas operator,in terms of the notion of holomorphic vector bundle. M.J.Cowen and R.G.Douglas,for the first time,applied the complex geometry into the research of operator theory.They have proved the Clabi Rigidity Theorem on the Grassman manifold,defined a new curvature function and indicated that this curvature is a complete unitary invariant of Cowen-Douglas operators.They conjectured that curvature is a complete similarity invariant of a Cowen-Douglas operators with index 1.But a counter example proved that the conjecture is wrong.It is these perfect results that have inspired people since 1997,to combine K-theory with complex geometry in order to seek the complete similarity invariants of Cowen-Douglas operators and their internal structures,we introduced three articles presented by Cao yang,Jiang Chunlan and others on this aspect.The Cowen-Douglas operators play an important role in studying the structure of non self-adjoint operators.Base on some of research on Cowen-Douglas operators,people have made headway in study of other operator classes.