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Báo cáo toán học: "Invariant subspaces of compact perturbations of linear operators in Banach spaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Subspaces bất biến của các nhiễu loạn nhỏ gọn của các nhà khai thác tuyến tính trong không gian Banach. | Copyright by INCREST 1979 J. OPERATOR THEORY 1 1979 225 - 260 INVARIANT SUBSPACES OF COMPACT PERTURBATIONS OF LINEAR OPERATORS IN BANACH SPACES LÁSZLÓ ZSID0 In 22 it was proved that if T is a self-adjoint operator on a complex Hilbert space H and K belongs to some Schatten-von Neumann ideal p of compact operators on H 1 p oo then T K has nontrivial invariant subspaces. This statement has been extended in 15 for T self-adjoint and K in the larger Macaev ideal of compact operators and in 12 for T normal the spectrum ff T of T contained in a simple c2 Jordan curve and K 6 ểa. The proofs of the above results are straightforward whenever a T K ơ T 0 so one may consider without loss of generality only the case a T K c a T . For T self-adjoint respectively unitary and 1 p oo with ơ T K . R respectively a T K on the unit circle I. Colojoara and c. Foias proved in 4 that T K is d-scalar for some admissible function algebra ã on ơ T K in particular T K is decomposable. This result is extended by M. Radjabalipour and H. Radjavi in 18 as follows if ơ T is contained in a closed c2 Jordan curve r 1 1 - 0-1 CO S1T d i r for Ầ ị r and ơ T K cz r then T K is decomposable. For a complex Banach space X the operator ideals analogous to p and may be defined in terms of the approximation numbers see 17 Section 8.1 . The main difficulty in extending the above results for operators on X consists in the absence of an appropriate determinant theory for operators on Banach spaces which would enable the estimate of the norm of the resolvent of T K in terms of the norm of the resolvent of T and the approximation numbers of K. The main result of this paper Theorem 3.1 consists just in a resolvent estimate of the above type. We reduce the general resolvent estimate problem to a finite-dimensional Hilbert space problem where the determinant theory is available. Once a convenient resolvent estimate is obtained for T K its decomposability follows by general criteria like 14 Th. 6 12 Th. 2.2 19 Th.