Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhiễu loạn phân tích của $ \ bar \ một phần $-nhà điều hành và các công thức đại diện không thể thiếu trong không gian Hilbert. | Copyright by INCREST 1979 J. OPERATOR THEORY 1 1979 187-205 ANALYTIC PERTURBATIONS OF THE Ớ-OPERATOR AND INTEGRAL REPRESENTATION FORMULAS IN HILBERT SPACES . VASILESCU 1. INTRODUCTION In this paper we present the construction of some operator-valued kernels which occur naturally in the study of certain integral representation formulas in particular in the analytic functional calculus for several commuting operators in Hilbert spaces. These integral kernels are obtained in connection with the analytic perturbations of a specific type of the 9-operator when d is regarded as a closed operator on Hilbert spaces of square integrable vector-valued exterior forms. Let H be a complex Hilbert space and H the set of all densely defined closed bounded operators acting in H. For any T G H we denote by @ T T tf T the domain of definition the range and the kernel of T respectively. In what follows we shall deal mainly with operators T G A i having the property âỉ T . l i l . roughly speaking with operators T satisfying T2 0. Such an operator T will be called exact when one actually has Ổ T T . The exactness of an operator T 6 ế with á T cz T is equivalent to the invertibility in H of the operator T T where T denotes the adjoint of T this is a simple and useful criterion from which some of the main results of this paper will be derived. Let us consider a finite system of indeterminates ơ ơx . . ơ . The exterior algebra over the complex field c generated by . ơ will be denoted by A ỚỊ-For any integer p 0 p n we denote by ApỊơ the space of all homogeneous exterior forms of degree p in O . ơ . The space A a has a natural structure of Hilbert space in which the elements A . A ƠJP 1 Ã . j n p 1 . . n as well as 1 G c d ơ form an orthogonal basis the symbol A stands for the exterior product . Let us define the operators Sjỉ ơj ỉ ỉ e A ữ j 1 .n . 188 . VASI1ESCU Then the adjoints of the operators are given by the formula 1-2 Sf j Ợ A t j J 1 . IÌ where c j ơj A Ẹ is
