tailieunhanh - Báo cáo toán học: On some contivution problems which are closely related to the theory of operations in spaces

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: | 1. OPERATOR THEORY 13 1985 299-417 Copyright by INCREST 1985 ON SOME CONTINUATION PROBLEMS WHICH ARE CLOSELY RELATED TO THE THEORY OF OPERATORS IN SPACES nx. IV CONTINUOUS ANALOGUES OF ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE WITH RESPECT TO AN INDEFINITE WEIGHT AND RELATED CONTINUATION PROBLEMS FOR SOME CLASSES OF FUNCTIONS MARK G. KREIN HEINZ LANGER In this paper the classes and SX a which were introduced in 1 2 play a fundamental role. Recall that a complex valued function h on 2a 2a is called Hermitian if h t h t 2a t 2a and that x o 0 a oo is the set of all continuous functions f g on 2a 2a which are Hermitian and for which the kernel Ff t s f t s Gg t s g t s g t - g s g 0 resp. 0 s t 2a has X negative squares. We put Tx spx oo x Sx oo. It turns out that an arbitrary function f e has a continuation in see 1 3 that is there exists a continuous Hermitian function f on the real axis such that f t f t 2a t 2a and that the kernels 7 . and F have the same number X of negative squares. This continuation can be uniquely determined or not and the question arises to give criteria for either case. If f e Px a has a unique continuation f e IPx we shall say that the continuation problem for f is determined otherwise it is called indetermined. In the second case the following problem naturally arises How to describe the totality of all the continuations f e sPx of The descriptions given in this paper will always be of the following form There exist four entire functions Wjk J k 1 2 such that the equality J w2ỵ z T z ww z 0 for some y 0 establishes a bijective correspondence between all continuations f e Px of f and all T e No. Here Nữ denotes the Nevanlinna class No consisting 300 MARK G. KREÍN HEINZ LANGER of all functions T which are holomorphic in the open upper half plane c map this half plane into itself and are extended to the lower half plane c_ by T z T z augmented by oo. The matrix function w Wyt i giving such a description will be called a resolvent matrix of .