Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các yếu tố của lý thuyết quang phổ từ tổng quát. | J. OPERATOR THEORY 3 1980 89-113 Copyright by INCREST 1980 ELEMENTS OF SPECTRAL THEORY FOR GENERALIZED DERIVATIONS L. A. FIALKOW 1. INTRODUCTION Let se denote an infinite dimensional complex Hilbert space and let SẾựTị denote the algebra of all bounded linear operators on . For A and B in let S AB or simply S denote the operator on defined by S Aii X AX XB. In the present note we characterize the Fredholm essential spectrum of S and we present several necessary or sufficient conditions for the range of S to be norm closed. In order to state our results in detail and place them in perspective we first recall some pertinent terminology and results from the literature. Let denote an infinite dimensional complex Banach space and let denote the algebra of all bounded linear operators on T. For T e S 3C letơíT ơi T and ffr T denote respectively the spectrum left spectrum and right spectrum of T. M. Rosenblum s fundamental result states that n c ff A ơ B s a p a e r A p e ơ ổ 22 and the identity ơ S ơ A ơ B is contained in 17 Theorem 10. Thus S is invertible if and only if A and B have disjoint spectra. Following 7 let ƠK T and ƠỖ T denote respectively the approximate point spectrum of T and the approximate defect spectrum of T . ƠÕ T Ấ 6 ơ T T Ấ is not surjective . For Te s cr 0-1 T 7it T and ffr T ơá T . In 7 c. Davis and p. Rosenthal characterized the approximate point and defect spectra of S as follows ơỏGB and Thus S is bounded below resp. surjective if and only if an A n ơâ B 0 resp. ơg A n ơn 5 0 . Subsequently the author proved that ơi S ơr S and ơn S ơị S 10 . Furthermore the range of S is norm dense in S Sf if and only 90 L. A. FIALKOW if ơrc A n ơle 5 0 and there exists no nonzero trace class operator X such that BX XA 11 Theorem see below for notation . For an operator T 6 let ker T and denote the kernel and range of T. Following 16 let nul T dim ker T and let def T dim i 7MT where Ỗữ Ty denotes the norm closure of. T . For X 6 let x denote the image of
