tailieunhanh - Báo cáo toán học: "On the reduction and triangularization of semigroups of operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Về việc giảm và triangularization của semigroups của các nhà khai thác. | J. OPERATOR THEORY 13 1985 63 71 Copyright by INCREST 1985 ON THE REDUCTION AND TRIANGULARIZATION OF SEMIGROUPS OF OPERATORS HEYDAR RADJAVI 1. INTRODUCTION Let if be a multiplicative semigroup of operators on a complex Hilbert space Jf. We shall be interested in sufficient conditions under which ỉf can be reduced . there exists a closed subspace Jt of Jf other than 0 and Jf which is invariant for every member of if. Some of these conditions are strong enough to give simultaneous triangularizability. This means the existence of a chain f of subspaces of J such that a f is maximal and b every member of f is invariant for if. The maximally requirement for V implies that if. is in f and if Ji_ is the closed linear span of .X e f . Jf c Ji then JI 9 JI_ has dimension 0 or 1. Kaplansky 4 gives triangularizability results for semigroups of operators on finite-dimensional spaces. One of his results is that if all the members of if have the same trace then Sf is triangularizable. It was conjectured in 7 that this should hold for trace-class operators on Jf. We prove it in Section 3 along with the infinite-dimensional analog of its companion in 4 if the spectrum ơ A of every A in if is a subset of 0 1 and the algebraic multiplicity of 1 in ơ X is r where r is a fixed nonnegative integer then ỉf is triangularizable. It should be noted here that Kaplansky s results are valid for very general fields. We also show the triangulariza-bility of a semigroup of compact not necessarily orthogonal projections. Other triangularizablity results for semigroups are given in 7 In Section 2 stronger results on reducibility are given. In particular it is proved that if tf is a semigroup of trace-class operators and if 1 is the unique element of maximal modulus in ơ A for every A in if then Sf has an invariant subspace. The algebraic multiplicity of 1 as an eigenvalue of A is allowed to vary even without bound. In what follows operator means bounded linear operator on Jf The notation Tfj is