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Báo cáo toán học: "The spectrum of Hilbert space semigroups "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các quang phổ của không gian Hilbert semigroups. | Copyright by INCREST 1983 J. OPERATOR THEORY 10 1983 87-94 THE SPECTRUM OF HILBERT SPACE SEMIGROUPS I. HERBST 1. INTRODUCTION Suppose P i t 0 is a strongly continuous semigroup of operators on a Hilbert space 2 f. We introduce the generator A of the semigroup by writing P i --- exp A and consider the problem of determining the spectrum ơ P z given some knowledge of the operator A. The inclusion 1.1 r exp tA exp ơ tA is known to hold for all such semigroups 5 but there are cases where the reverse inclusion fails 5 . Here exp ơ jA is the closure of the set e z z 6 ơ tA . In fact as is demonstrated in 5 it is possible to have ơ X 0 while exp tA has circles in its spectrum. There are of course examples where o A 0 while ơ exp tA 0 for t 0 so that in general ơ exp tA is not determined by ơ A alone. It follows from the assumed strong continuity of P i that the bound 1.2 P í À exp ícu t 0 is valid for someẨ 1 and eR 5 If Kcan be set equal to one then P z exp tco -- exp t A co is a strongly continuous semigroup of contractions. In Hilbert space such semigroups are well studied and structure theorems comparable to the spectral theorem for normal operators are known 1 . Using this additional structure L. Gearhart 2 showed that if K 1 in 1.2 and y is separable then ơ P í can be determined from a knowledge of a A and the behavior of z X -1 for z near infinity. Specifically Gearhart proved the following result 2 In stating the result we use the notation p B for the resolvent set of an operator B. Theorem 1.1. 2 . Suppose exp tA -. Z 0 is a strongly continuous semiroup of operators on a separable Hilbert space with IIexp tA II exp ico for some co e R. Then 88 I. HERBST 1 exp z0 e p exp AỴ if and only if z0 2nin e p A for all integers n and sup j z0 -r 271ÙÍ 00 Ỉ neZ 2 0 ep exp A if and only if there are numbers c0 and 0 0 0 so that z Re z co0 c p A and for Re z a0 il z - c0 Rez - Gearhart s proof of this theorem is not elementary. It is the purpose of this note to give a .