Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Phân phối bán nhóm các nhà khai thác dưới nhiệt độ. | J. OPERATOR THEORY 5 1981 47 52 Copyright by INCREST 1981 ON DISTRIBUTION SEMI-GROUPS OF SUBNORMAL OPERATORS IOANA CIORĂNESCU 1. INTRODUCTION Let X be a Banach space and A a closed densely defined operator in X with domain DỊAỵ an AQ-valued distribution Ỗ with the support contained in 0 -Ị-OO is said to be a regular distribution semi-group . in short of generator A if ể and A satisfy the equations A d dt Ỗ X Ix and à ảt Ỗ ID A . An . s is said to be an exponential distribution semi-group of type si Cử . in short if g satisfies the following condition there exists a real a such that is an X X -valued tempered distribution for any g a . Distribution semi-groups were defined and studied by J. L. Lions in 7 . oo Let us denote Y D An and endowe Y with the Frechet topology deter-n l mined by the norms y TLv . 7 0 Then the following conditions on the operator A are .equivalent i A is the generator of a . ii the resolvent R Ấ A exists in a logarithmic region of the form A ẲeC ReẤ a logịlmẤI p Re Ẳ where a p 0 y e R are some given constants and satisfies p f beeing a polynom with positive coefficients. iii the resolvent set p A is not empty and the restriction of A to Y A Y is the generator of a locally-equicontinuous semi-group of class Co in Y. The equivalence i ii was proved by J. Chazarain in 2 and the equivalence i iii was obtained by T. Ushijima in 11 48 IOANA CrORĂNESCU The . s and the semi-group CZf 0 generated by Ay may be expressed in terms of the resolvent R f A as follows f p c p 2 R z A dẰ for each cpeS dA where Q is the space of all indefinitely differentiable functions on the real line with compact support p 2 eA ợ t dí and dA is the boundary of A Jr Ifx lim Ự hA itlhix for each X e Y where the convergence is uniform with respect to t in every finite interval in 0 oo see 8 and 9 . Moreover we recall that S cp p t U xdt for each X G Y ipeS Jr see 11 and that the semi-group property is given by ố ợ í Ế p ó ý .
