TAILIEUCHUNG - Báo cáo toán học: "Operator decomposability and weakly continuous representations of locally compact abelian groups "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Operator decomposability và yếu liên tục của đại diện có sẵn tại địa phương nhỏ gọn nhóm abelian. | J. OPERATOR THEORY 7 1982 201-208 Copyright by INCREST 1982 OPERATOR DECOMPOSABILITY AND WEAKLY CONTINUOUS REPRESENTATIONS OF LOCALLY COMPACT ABELIAN GROUPS JỠRG ESCHMEIER 0. INTRODUCTION Let G be a locally compact abelian group r its dual group U G the space of complex valued functions on G integrable with respect to Haar measure and M G the Banach algebra of regular complex Borel measures on G. We show Corollary 3 that for non-discrete G there exists a measure ỊẦ e M G such that the convolution operator T L G - L GỴ gn g is not decomposable thus answering in the negative a question of I. Colojoara and c. Foias 5 p. 218 . On the other hand for any measure ỊẢ 6 Ma Md the subalgebra of M G consisting of all measures Ằ whose continuous part jda belongs to L G the operator Tfl is strongly decomposable. More generally using recent results of D Antoni R. Longo and L. Zsido 6 we obtain that for any weakly continuous representation u of G by isometries on a Banach space X and for any Jtz e Ma Md the generalized convolution operator n i e B X defined by n fi U s dfi s G is strongly decomposable. For the notion of decomposable and strongly decomposable operators we refer the reader to 3 5 7 . I want to thank Professor L. Zsido for inspiring this work. 1. THE COUNTEREXAMPLE Let X be a Banach space and B X the space of continuous linear operators in X. For any T e B X we denote by a T the spectrum of T and by Inv T the system of all closed linear T-in variant subspaces of X. We say an operator T e B X 202 JỠRG ESCÍỈME1ER has the weak 2-SDP Spectral Decomposition Property if for any open covering ơ T c u u2 there are spaces X1 x e Inv 7 such that X Xj x2 a T Xfi c u 1 2. Lemma 1. Let X Y be Banach spaces Te B X s e B Y and A e BỌỉ Y injective such that AT -- SA. ỉf T has the weak 2-SDP we have ơ T c o S . Proof. Let T have the weak 2-SDP. For z0 ị ơ S we can find Xj X. e Inv T such that X- xj x2 o-fr Xj p S ff T X2 cC z0 . if r is a cycle surrounding a S in pỢLXỵ such that ơ S

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