tailieunhanh - Báo cáo toán học: "Functional calculus and invariant subspaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Chức năng tính toán và subspaces bất biến. | Copyright by INCREST 1980 J. OPERATOR THEORY 4 1980 159-190 FUNCTIONAL CALCULUS AND INVARIANT SUBSPACES c. APOSTOL INTRODUCTION This paper is an attempt to generalize the results of J. Agler 1 s. Brown 4 s. Brown B. Chevreau and c. Pearcy 5 and J. G. Stampfli 27 on invariant subspaces. We shall show that the techniques of s. Brown can be used to produce invariant subspaces for polynomially bounded Hi-tuples of operators acting in Banach spaces. We have to say that our results are not complete as in the above quoted papers except for m 1 in particular cases because of the difficulties of spectral nature for m 1 and of the imperfection of general Banach spaces. The paper is divided in five sections. In 1 and 2 we develop an func- tional calculus for a polynomially bounded Hi-tuple A e where ỂÍ denotes a complex separable Banach space. If the approximate point spectrum of A is enough rich then the D functional calculus becomes a weak homeomorphism between and a weak closed subspace of áf regarded as the dual of a tensor product space. Thus we can speak about the weak closure of the algebra generated by A. In 3 we produce hyperinvariant subspaces only for the reductions we shall need in the sequel. In 4 we produce invariant subspaces for A in case it has rich approximate point spectrum and 3C has an unconditional basis or an unconditional finite-dimensional decomposition determined by some compact injective scalar operator. Theorem is a direct correspondent of the main result of 5 . In 5 we release the hypothesis on S and our results involve only restrictions to invariant subspaces of Hi-tuples having functional calculus with continuous functions on D . quasiscalar m-tuples . Sample results in this section are If ỵ is reflexive T e is subscalar and ỠD c r T cz D then T has a proper invariant subspace Theorem . The restriction to an invariant subspace of the multiplication with the argument in p h 1 p oo where m denotes a finite positive Borel measure in c