TAILIEUCHUNG - Báo cáo toán học: "On equivalence of Toeplitz operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nó tương đương của các nhà khai thác Toeplitz. | J. OPERATOR THEORY 7 1982 107-172 Copyright by INCREST 1982 ON EQUIVALENCE OF TOEPLITZ OPERATORS CARL c. COWEN Dedicated to Professor Paul R. Halmos on his 65 th birthday. I. INTRODUCTION For f in of the unit circle cD the Toeplitz operator Tf is the operator on the Hilbert space H2 of the unit disk D given by Tfh Pfh where p is the orthogonal projection of EfcD onto H2. As usual we identify H2 as a subspace 00 co of LfcD by y a z y a è where y a 2 oo. Toeplitz operators have been ir-0 n-0 Studied widely for the past thirty years but with two major exceptions little seems to be known about the similarity or unitary equivalence of Toeplitz operators. The first of these exceptions is the computation of spectral measure for self-adjoint Toeplitz operators 12 13 9 10 11 . The other exception is the striking work of Clark and Morrel on the similarity of Toeplitz operators with rational symbol 3 4 5 6 . Tn this paper we give an easy sufficient condition for unitary equivalence appeal to the work of Rosenblum to show that it is not necessary and prove a strong converse for some cases in which the symbol f is in TP . We recall that u is an innerfunction if u is in ỉ2and w eie Ị i lim u rei0 1 r- i - for almost all 0. Hoffman s book 8 is a good reference for Hardy spaces and factorization theory. If It is an inner function we say the order of u is II if u is a finite Blaschke product of order 7 otherwise we say the order of It is infinity. We will show Theorem 1 that if h is in ư-yõD and II is inner then T u Th where n JI order II possibly infinity . We will write for unitary equivalence and for similarity. It follows that if u and V are inner functions of the same order then Th-1 T i . If ỈP is a set of operators let ÍP denote the set of operators that commute with everything in ÍP and ỈP fry. We will also show Theorem 2 that if f and g are in 7 such that for some u and V inner ĩ T and Tg Tv then Tf s Tg if and only if Tf Tg if and only if there is h in J and inner .

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• báo cáo của tạp chí Journal of Operator Theory
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