TAILIEUCHUNG - Báo cáo toán học: "On eigenvalues in the essential spectrum of a Toeplitz operator "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên các giá trị riêng trong quang phổ cơ bản của một nhà điều hành Toeplitz. | J. OPERATOR THEORY 13 1985 151-162 Copyright by INCREST 1985 ON EIGENVALUES IN THE ESSENTIAL SPECTRUM OF A TOEPLITZ OPERATOR p. R. AHERN and D. N. CLARK INTRODUCTION Let T be the unit circle in the complex plane c. For a bounded measurable function p on T define the Toeplitz operator Tv on H2 of the unit disk by v p p where p is the projection of L2 on H2. If p is continuous the Fredholm theory of 7 is well known 6 Chapter 7 . Indeed pe Tj the Fredholm resolvent set is the complement of the curve p T and the index of Ty ư for 2 ị p T is a p 2 the negative of the winding number of the curve p T about 2. In addition the index o p 2 is equal to the dimension of the kernel ker 7 2Z if co p 2 0 and m p 2 dim kerCT 2z if U J9 2 0. The present paper is a report on an investigation of the dimension of ker T 2Z when f is continuous and 2 lies on the curve p T so that Tv 2Z is not a Fredholm operator. Some previous work on the eigenspaces of non-Fredholm Toeplitz operators may be found in 10 5 3 8 . Our work differs from that of these otherauthors in that we seek a description of eigendimension in terms of the geometrical properties of the curve p T modeled as closely as possible on the winding number characterization of dimker L 2Z for 2 ị p T . A test question for a geo-metricaltheory of eigendimension is the problem of whether dim ker 7 2Z 0 can hold for 2 e dff Tp the boundary of the spectrum of Tv 2 9 . For our formula we assume p e has the form 1 p e 9 JJ e-i e ie a z ei j-1 where otj 0 j 1 . n and h is continuous and nonvanishing on T. In Part 1 we prove that y z 0 which determines dim ker Tv is characterized as follows to 152 p. R. AHERN and D. N. CLARK be made precise in Section . Let 2 be a connected component of C ự T with 0 G dQ. If the boundary of Q has positive inner angle at 0 we label as negative the arcs f t p eu 0j t Oj È such that the two arcs of .j itself meet at positive angle at 0 and Í2 lies on the right as Ảj is traversed through a neighborhood of

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