TAILIEUCHUNG - Báo cáo toán học: "Hypercontractions and subnormality "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hypercontractions and subnormality . | Copyright by INCREST 1985 J. OPERATOR THEORY 13 1985 203- 217 HYPERCONTRACTIONS AND SUBNORMALITY JIM AGLER 0. INTRODUCTION In this paper will always refer to a separable Hilbert space over the complex numbers. will denote the bounded linear transformations of . The following theorem is well known 4 9 . Theorem a. Let s denote the unilateral unweighted shift of multiplicity one and let denote the direct sum of a countably infinite number of copies of s . Let T e LdfMLf T has an extension to s oo if and only if T 1 and Tm - 0 strongly as m oo. In 2 the following analog of Theorem A with s replaced by B the Bergman shift was proved. Theorem B. Let T e T has an extension to if and only if T 1 1 2T T T 2T2 5 0 and Tm 0 strongly as m -too. A comparison of Theorem A and Theorem B above seems to suggest that they are but the first two theorems in an infinite sequence of theorems. This sequence of theorems is described in Section 1 of this paper. Loosely speaking for each positive integer o I we introduce a class of operators T which is defined in a simple way using a system of polynomial inequalities in T and T and which we call the n-hypercontractions for each n there is also a weighted shift Sn. Our theorem Theorem then states that T e has extension to S 0O if and only if T is an n-hypercontraction. When n 1 the theorem is Theorem A above and when n 2 it reduces to Theorem B above. Our initial proof of Theorem which appears in Section 1 is based on the techniques in 2 In Section 2 we give a different proof based on the Rovnyak--deBranges construction 7 and 4 . While considerably less amenable to generalization than the proof in Section 1 the proof in Section 2 has the advantage of being vastly more elementary. In Section 3 we answer the question What are the operators that are n-hy-percontractive for every nl The answer Theorem is surprising not only because 204 JIM AGLER it is SO simple T is an n-hypercontraction for every n if and only if T is a .

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