TAILIEUCHUNG - Báo cáo toán học: "Similarity of smooth Toeplitz operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tương tự của các nhà khai thác Toeplitz mịn. | J. OPERATOR THEORY 12 1984 319-329 Copyright by INCREST 1984 SIMILARITY OF SMOOTH TOEPLITZ OPERATORS DERM1NG WANG I. INTRODUCTION Let T denote the unit circle z 1 in the complex plane c and dm z l 27t dr normalized Lebesgue measure on T. Let 2 be the Lebesgue space of equivalence classes of square integrable with respect to dm z functions on T and H2 the L2-closure of polynomials. For a bounded dm z -measurable function g the associated Toeplitz operator TS H2 - H- is defined by Tgh P gh where p is the orthogonal projection of 2 onto H2. The function g is called the symbol of Tg. In this paper we shall be interested in the case when the symbol is smooth and its negative Fourier coefficients decay exponentially. More precisely let J 1 n ỷ 1 oo denote the set of all functions g e C n T with Fourier series g eiz ữkeikl k oo satisfying a_t crk for k 1 2 3 . and positive constants c and r r 1. It is clear that c J m if n m. We obtain the following Theorem 1. If F e J 1 F is one-to-one and F never vanishes on T then Tp does not have an eigenvalue on the boundary of the spectrum ơ Tp of Tp. If in addition t - F eiz is orientation preserving then Tp has no eigenvalues. Theorem 2. If Fe Jw F is one-to-one F never vanishes on T and t - F eiz is orientation preserving then Tp is similar to an analytic Toeplitz operator Tt where T is a Riemann mapping function of jz 1 onto the interior of the curve F T . Note that with proper orientation every function analytic and one-to-one in a neighborhood of T satisfies the hypotheses of the above two theorems. The origin of our similarity theorem Theorem 2 dates bach to a paper of p. L. Duren 9 . There Duren obtained Corollary 1 of Section 4 below under the assumption that F z j z y z with p y . In her dissertation 10 J. H. Morrel proved Theorem 2 for F a trigonometric polynomial. Subsequently D. N. Clark and J. H. Morrel 5 obtained the same similarity theorem under the assumptions that Fis rational with poles offT and is one-to-one on .

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