Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Thuộc tính quang phổ của số nhân tổng quát. | g Copyright by INCREST 1985 J. OPERATOR THEORY 14 1985 . 277 - 289 SPECTRAL PROPERTIES OF GENERALIZED MULTIPLIERS FLORIN RADULESCU In the present paper we shall be concerned with an extension of the commutator and the multiplier of two operators. Namely given s T acting on Banach spaces X Y and 0 an analytic complex valued function defined in a neighbourhood of the Cartesian product ơ S Xơ T we define an operator 0 S T L Y X - L Y X so that for simple functions such as 0 z w z IV or 0 z iv ZIV we obtain the commutator and the multiplier of 5 T . the operators C S T V SV - VT and M S T V SVT respectively . . We shall prove directly that the mapping 0 - Ỡ S T from the algebra of germs of analytic functions in neighbourhoods of a S xa T into L L Y X is a morphism of algebras and using this result we shall deduce an evaluation for the spectrum of ỹ S T see also 3 5 . Following the ideas from 4 we shall obtain some spectral properties of 0 .S T the main result is Theorem 10 which is an extension of the results obtained in 4 . For definitions and main techniques used in this paper we refer to 8 . 1. DEFINITIONS AND GENERAL RESULTS Let X Y be two Banach spaces and let 5 e L X T G L y be two bounded operators. Let Ds DT be two open sets containing ơ S and ff T respectively. Let 0 DsxDT - c be an analytic function. If rs rT are two regular contours contained in Ds and DT surrounding ff S and a T respectively we define 0 z T 27TÍ -1 ị 0 z w w - T -1 dw. rT For V belonging to L Y X we define 0 S T V 2m -1 z - S -1KỚ z T dz. rs 278 FLORIN RÃDƯLESCL It is obvious that 0 S T is a bounded operator from L Y X into L Y X which does not depend on the particular choice of rs and of rT. If 0 has the par ticular form 8 z w -- ỵ z g . n where fj gj are analytic functions on Ds and DT respectively then using the ana-lytic functional calculus we obtain 0 S T V By putting ỡ z w z w or 0 z w zw we obtain the commutator and the multiplier. Lemma 1. Let O D T - c be another analytic
