TAILIEUCHUNG - Báo cáo toán học: "Estimates of functions of power bounded operators on Hilbert spaces "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Ước tính của các chức năng của quyền lực giới hạn các nhà khai thác không gian Hilbert. | J. OPERATOR THEORY 7 1982 341-372 Copyright by INCREST 1982 ESTIMATES OF FUNCTIONS OF POWER BOUNDED OPERATORS ON HILBERT SPACES VLADIMIR. V. PELLER I. INTRODUCTION One of the basic methods in the spectral theory of operators is the construction of a rich functional calculus for the class of operators under investigation. The main difficulty to construct such a calculus is to obtain sharp estimates of norms of functions of operators. The most famous and important inequality of this type is J. von Neumann s inequality 14 __ def ll p uil sup p 0 Ị c e D IMIWOO for any contraction T . II T 1 on a Hilbert space and for any complex polynomial p. The main problem treated in this paper is the investigation of the class of power bounded operators on a Hilbert space . such operators T that II T const n 0. In other words the problem is to calculate explicitly the norm III Hie - sup p T ll T c n 0 c 1 on the set of polynomials. If T is an invertible operator with T const n e z it follows from s theorem 18 that T is similar to a unitary operator and thus for any trigonometric polynomial f the following inequality holds MU II const MIloo- However s. R. Foguel 8 constructed an operator T on a Hilbert space such that IIT II const n 0 but T is not similar to a contraction. This operator is defined on 2 2 by the operator matrix T s lo SỴ 11 - 1789 342 VLADIMIR V. FELLER where 5 is the shift operator on Í2 S x0 A j . 0 x0 xk . and Q is the orthogonal projection onto the subspace of if2 spanned by ek k 3n neZ . A. Lebow 11 showed that this operator is not polynomially bounded . for any k Q there exists a polynomial p such that MD i k p oo-n Besides the power bounded operators we consider the operators with the growth of powers of order a a 0 . operators T satisfying ll T const l na n e z . It is evident that for any operator T on a Banach space such that 7 i const l n a zieZ the following inequality holds 1 II P T II const ỵ ị Ặ n 1 -r ti x peẩ A íỹA stands for

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