TAILIEUCHUNG - Báo cáo toán học: "Characterization of some Harnack parts of contractions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đặc tính của một số bộ phận Harnack bóp . | J. OPERATOR THEORY 2 1979 233-245 Copyright by INCREST 1979 CHARACTERIZATION OF SOME HARNACK PARTS OF CONTRACTIONS T. ANDO I. SUCĨU and D. TIMOTIN In 1 Zoia Ceausescu introduced a relation of domination between contractions acting on a Hilbert space characterized by some simple norm inequalities-This relation is easily shown to be weaker than Harnack domination cf. 4 5 for the latter . Zoia Ceausescu asked whether the equivalence classes of this relation do coincide with Harnack parts. In this paper we give a positive answer to this question in the case when the contractions have spectrum contained in the open unit disc of the complex plane. As a consequence we prove that the Harnack part of any spectral norm strict contraction is not trivial. For the general case we show that the answer is negative by exhibiting a class of counterexamples. The techniques used yield also some other results concerning Harnack domination and its connection to the relation introduced by Zoia Ceausescu. 1. NOTATION AND PRELIMINARIES. In the sequel T T will be linear contractive operators acting on the Hilbert space 2 u acting on X and U acting on X will be the minimal isometric dilations of T T respectively. We shall denote by r T the spectral radius of T and by TẰ the operator defined by 2 1 We say cf. 4 that T is Harnack dominated by T notation T - T if there exists a positive constant a such that for any analytic polynomial p verifying Re p z 0 for I z I 1 we have Re p T a Re p T . H jy Obviously - is a preorder relation and we shall denote by the equivalence relation induced by it. The equivalence classes are called Harnack parts. H It was shown in 5 that T - T is equivalent to the following property 234 T. ANDO I. suctu and D. TIMOTIN There exists a bounded operator S X - X such that for any hữ hi . . . ỉ eX we have s Ẻ Ẻ uJhJ- 7 0 J o Since X f IPPC f U J PC it is obvious that in this case s is the 7 0 i 0 unique bounded operator from X into X which intertwines U and u and .

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