tailieunhanh - Báo cáo nghiên cứu khoa học: "Perturbations of nest algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhiễu loạn của đại số tổ. | J. OPERATOR THEORY 1 1979 137-150 Copyright by INCREST 1979 PERTURBATIONS OF NEST ALGEBRAS THOMAS FALL WILLIAM ARVESON and PAUL MUHLY INTRODUCTION Let Pn n 1 be a sequence of finite dimensional projections on a Hilbert space ff such that Pn t 1. An operator T on is said to be triangular resp. quasi-triangular relative to p if P TP 0 for each n 1 resp. IIPi TP - 0 as n - oo . Let and denote the corresponding two algebras of operators on iff. The algebra was introduced in 1 where it was shown that every operator in is a compact perturbation of an operator in .Ỉ. One deduces easily from this that SLff J -ff which can be regarded as a characterization of the compact perturbations of operators in here ff denotes the algebra of all compact operators on ff . It is natural to ask the extent to which a characterization like this is valid for other nest algebras in place of J. Specifically consider the triangular algebra J O 1 of the unit interval defined as the set of all operators on L2 0 1 which leave invariant each subspace L2 0 t 0 t 1 What we seek is a characterization of operators in y 0 1 ff in terms of the projections p -. 0 t 1 where pt denotes the projection onto L2 0 i . This is accomplished in section 2 Corollary of following a rather general discussion in section 1 which implies thatyjo 1 ff is norm-closed. In fact we present a characterization of compact perturbations of arbitrary nest algebras which covers both cases 0 1 and at once. 1. CLOSURE PROPERTIES OF PERTURBED ALGEBRAS It is easy to see that for any c -algebra d of operators on a Hilbert space the algebra sd- -ff of all compact perturbations of operators in is norm-closed. Indeed ff 7t-1 7l where n is the Calkin map and 7i j is closed since 7Ĩ is a c -homomorphism 5 . This is false for general norm closed non-self-adjoint operator algebras 3 138 T. FALL w. ARVESON and p. MUHLY In this section we obtain a result which contains the required information about nest algebras as well as a much