TAILIEUCHUNG - Báo cáo toán học: "Geometrical means of eigenvalues "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hình học có nghĩa là phân nhóm của các giá trị riêng. | J. OPERATOR THEORY 7 1982 173-178 Copyright by INCREST 1982 GEOMETRICAL MEANS OF EIGENVALUES ENDRE MAKAI Jr. and JAROSLAV ZEMANEK I. INTRODUCTION Let T be a bounded linear operator on a complex Banach space X. The spectrum of T will be denoted by a T and the spectral radius by T . By the essential spectrum we mean the spectrum of T modulo the compact operators. The essential spectral radius cuts the spectrum ơ .T into two parts. We shall be interested in the outer part A T p. e o T m I nJ. It is well-known that this set is at most countable and consists of isolated eigenvalues of finite multiplicity. We refer to the introduction of the paper 6 for a more thorough explanation of these facts. So we can denote the eigenvalues in A T in such away that z X T z where each eigenvalue is counted according to its multiplicity which is the dimension of the range of the corresponding spectral projection . If there are only n 0 1 2 . such eigenvalues including multiplicities we put formally IP 1 UI I4 2 UI I To- Let u be the unit ball of X. For each n 1 2 . let e T be the infimum of all e 0 such that the set T U can be covered by n balls of radius a with centres arbitrary in X. These are the entropy numbers of A. Pietsch 3 p. 168. It was noted in 6 Corollary that for each fixed n 1 2 . it holds 1 T ff Jim e TN llN. N- x Recently B. Carl 1 has defined gn T inf k -n ek T and called it the n-th k entropy modulus of T. A natural geometric interpretation of these quantities comes from considering the n-dimensional complex volume of the coverings of the set ĩ 74 ENDRE MAKAI Jr. and JAROSLAV ZEMANEK Ĩ U by finite number of balls of equal radii. He also established in 1 some algebraic properties of these quantities among them the submultiplicativity property g ST gn S g T valid for each n 1 2 . and every s T this is easily verified directly from the definition of g . It is then a matter of algebraic calculation to see that the limits Gn T - lim g T y N N- X exist for each fixed n

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