TAILIEUCHUNG - Báo cáo toán học: "Spectral properties of $L^p$ translations "

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 $ L ^ p $ dịch. | J. OPERATOR THEORY 14 1985 87-111 Copyright by INCREST 1985 SPECTRAL PROPERTIES OF Ư TRANSLATIONS . GAUDRY and w. RiCKER 0. INTRODUCTION In 6 u. Fixman showed that the bilateral unit shift in P Z where 1 p GO p 2 and z is the additive group of integers is not a spectral operator in the sense ofN. Dunford 3 This example is in fact typical of translation operators in general locally compact abelian groups. For if G is such a group then except in trivial cases translations in LP G I p GO p 2 are not spectral see 9 and 2 Chapter 20 for example. However it is natural to expect translations to be spectral in some sense because they are isometries in LP G and hence analogues of unitary operators. This point was taken up in 9 and also in the recent article 1 4 where it is shown that translations can indeed be expressed in the form 1 e dỔ À R where Ổ Ấ 2. e R is an associated spectral family of commuting projection operators satisfying certain properties 1 4 and the integral 1 which can be interpreted as being over the unit circle T of the complex plane c exists in a certain well defined sense. But it should be stressed that in general the spectral family does not generate a ơ-additive projection-valued spectral measure. However as suggested in the note 18 an alternative interpretation of 1 is possible. Namely an operator may fail to be spectral solely because its domain space is too small to accommodate the projections needed to form its resolution of the identity. Accordingly if interpreted as acting in a suitable space containing the domain space it happens often that such an operator is spectral in the sense of Dunford. This has the advantage that the operator then has associated with it a rich functional calculus. It is shown in 18 Example that the bilateral unit shift in P Z 1 p 2 which has spectrum equal to T is a scalar-type spectral operator in this wider sense with resolution of the identity supported in T. Earl Berkson posed the natural 88 G. I. GAUDRY and

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