Banach Reducibility Of Operator Weighted Shifts

Lot H denote the complex separable Hilbert space. 2(H)=i=0xH. if {Wi}i=1+x is a sequence of uniformly bounded linear operators on H .S 6 (2(H)) , andthen S is called a unilateral operator weighted shift,denoted by S ~ {Wi} . the set of all this kind of operators is denoted by IW2(H).Particularly,let C be the complex plane.Cn - k=1n, C. 2(Cn) = Cn.then S is called a n-multiple unilateral operator weighted shift . the set of all this kind of operators is denoted by IW2(Cn).Operator weighted shifts form an important class of operators that people are interested in . One pay more attention to them because they are often used to make examples and counter examples, moreover they are closely related with some general problems in operator theory .Let T (H).M LatT. if there exists a N Latr.such that M N = {0} and M + N = H.then we call .M a Banach reducible subspace of T. T is said to be Banach reducible if there exist a nontrivial Banach reducible subspace,if not T is said to be Banach irreducible. T is Banach reducible if and only if there exists a nontrivial idempotent operator that commutes with it, if and only if T is similar to a reducible operator.The problem of operator redudbility pay a significant role in operator theory.When H is a finitely dimensional space, T is strongly irreducible.that is Banach irreducible, if and only if it can be represented as a Jordan block under some OXB,so strongly irreducible operator is a natural generalization in infinitely dimensional space.This idea has been demenstrated being intelligent by Zejian Jiang .Chunlan Jiang and their partners.The forward unilateral scalar weighted shift is strongly irreducible. In[1], Juexian Li proved that if S IW2(Cn). and e(T) is not connected .then Sis Banach redudble.Does this property hold for more extensive operators in IW('2(H) ? In this paper, we consider this question and give a positive answer. First.for IW 2(H) we prove the following lemmaLemma 2.1 For eachHi, there exists an OXB { } . such that every Wi can be represented asThus, every S IW2(H) is unitarily equivalent to an upper triangular unilateral operator weighted shift.Let V = { : r1 < | | < r2 } F(S) be a hole of (S).hx 0 V.we can obtain an n dimensional OXB of (Ran/(S -0)) restricting on H.Every S ~ {Wi} can be represented as an upper triangular unilateral operator weighted shift under this OXB where every IF, has the form of (1) .Then we can show the followingLemma 2.3 Let S IW2(H).S ~ {Wi}. then S is unitarily equivalent towhere 2(M)).B 2(H M)) are upper triangular unilateral operator weighted shifts with invertible multiplicity.Then by analyzing the relations of position among A0 and .4. B. we obtain.Proposition 3.1 S IW2(H). S ~ {Wi). 0 V. S is unitarily equivalent to the form of (2) .then 0 (B) = (B). 0 > r(B).From lemma 2.3. we may let A IW72(Cn).for the operators in IW72(Cn).we get to the following propositions from some conclusions in [l].Proposition 3.2 5 IW2(H). S ~ {W'i} . A0 V. S is unitarily equivalent to (2). then A0 1(A). 0 < r1(A).For A (H1). B (H2).Rosenblum operator AB is defined on (H2, H1) as TAB(X) = AX - XB. for any A (H2,H1). By using this operator we obtainLemma3.6 .A if 1(A) r(B] = o. then C RanTAB of each operator C of the followingformFrom lemma3.6. we can find an invortible operator such that an operator in IW2(H) is similar to a reducible operator, so we have the main theoremtheorem 3.1 S IW2(H). S ~ {Wi} . if e(5) is not comected.then 5 is Banach reducible.In this paper, we also get some conclusions about the Cowen-Douglas operators, let be a connected open subset of C. Bn( ) denoted the set of operators B in (H) satisfying(a) (B):(b)Ran(B -) = H. :(d) dimker(B - ) = . . Then call an operator in Bn( ) a Cowen-Douglas operator.Corollary3.1 S IW 2(H)..S - {Wi}+ - 0 V. 5 is Militarily equivalent to (2).then A Bn( ).where n is the negative vahie of iucI(S - 0).Analogously. we define a new concept Bx( ).tlie following remainedquestion (1) for 5" G I\V(2(H).th...