Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Co semigroups và quang phổ của $ A_1 \ Otim tôi + I \ $ A_2 Otim. | J. OPERATOR THEORY 7 1982 61 - 78 Copyright by INCREST 1982 CONTRACTION SEMIGROUPS AND THE SPECTRUM OF Al I I A2 IRA w. HERBST 1. INTRODUCTION In the following we will be mostly concerned with a strongly continuous semigroup of operators Pit 0 on a separable Hilbert space satisfying 1-1 P 0l e 1 . We introduce the generator A of P t by writing P t e- A. We will analyze the relationship of the spectrum of the operator P t to that of A and to the behavior of the resolvent z A -1 as a function of z. The spectral mapping equation ơ e- J 0 e_ ơ 4 t 0 is known to hold for a large class of A. 3 Here a B is the spectrum of B and e íơ J e z z e ơ f . For example if P t is a holomorphic semigroup is known to be valid 3 . While the inclusion ơ e-M 2 e- ơM is always true 3 can fail in a dramatic way. In fact Hille and Phillips 3 have constructed an operator A with e-M satisfying such that ơ A is empty while e-M has circles in its spectrum. In a recent paper 2 Gearhart has related the spectrum of an operator e A satisfying our assumptions to the behavior of z A -1 when z is near infinity. Explicitly Gearhart shows among other results 2 62 IRA w. HERBST Theorem . Suppose e- 0Ị is a strongly continuous semigroup of operators on a separable Hilbert space satisfying . Then e - is in the resolvent set of e A if and only if z0 -Ạ 2nin is in the resolvent set of A for all integers n and sup s0 2t0h A 1 co . ez In the next section we make use of Gearhart s theorem to show that if fails it must fail rather dramatically. If Al and A2 are generators of holomorphic semigroups on Hilbert spaces and . 4 respectively e- 4 e Ai r 0 is also a holomorphic semigroup on . 4 We denote its generator by A A 14-1 A2. The spectral mapping formula is correct 3 for generators of holomorphic semigroups so that using ơ C D ơ C ơ Đ which holds for arbitrary bounded operators c and D we have for t 0 Ị 41 e-ic .4 _ e This leads by a simple argument 6 to a theorem of
