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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hình ảnh quang phổ của các chức năng của nhà khai thác. | J. OPERATOR theory 8 1982 . 391-400 Copyright by INCREST 1982 SPECTRAL PICTURES OF FUNCTIONS OF OPERATORS c. BOSCH c. HERNANDEZ E. De OTEYZA and c. PEARCY 1. INTRODUCTION Let ye be a separable infinite dimensional complex Hilbert space and let yyye denote the algebra of all bounded linear operators on ye. The concept of the spectral picture of an operator T in ỈỂ ye was originated by one of the authors in 9 and was subsequently studied by Chevreau 4 . The notion seems to be a useful one. In particular it enables succinct statements to be given of two recent major theorems in operator theory. Thus the theorem of Brown-Douglas-Fillmore Ị3 characterizing essentially normal operators up to compalence becomes Two essentially normal operators in yỵye are compalent if and only if they have the same spectral picture. Furthermore the Romanian characterization of quasitrian-gular operators 1 can be stated thus An operator in yyye is quasitriangular if and only if its spectral picture contains no negative number. See 9 for definitions. In this note we begin a program of calculating the spectral picture of various constructs of an operator T in yyye in terms of the spectral picture of T. In particular we completely determine the spectral pictures of all operators of the form 7 where f is a function analytic on an open set containing the spectrum of T. We then apply these results to give some very short proofs of various facts about quasitriangular operators. First some notation and terminology that will be needed later are introduced and the relevant facts about spectral pictures are reviewed. We denote by K the ideal of compact operators in yyye and by 7Ĩ the quotient map of y ye onto the Calkin algebra The spectrum of an operator T in ee yey will be denoted as usual by ơ T and the essential spectrum of T i.e. the spectrum of n T in the Calkin algebra by ƠC T . Similarly the left and right essential spectra of T notation ơle T and ơre T are the left and right spectra of tt T
