A characterization of k-hyponormality via weak subnormality |
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Authors: | Raúl E Curto Il Bong Jung Sang Soo Park |
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Institution: | a Department of Mathematics, The University of Iowa, College of Liberal Arts and Sciences, Iowa City, IA 52242, USA b Department of Mathematics, College of Natural Sciences, Kyungpook National University, Taegu 702-701, Republic of Korea |
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Abstract: | An operator T acting on a Hilbert space is said to be weakly subnormal if there exists an extension acting on such that for all . When such partially normal extensions exist, we denote by m.p.n.e.(T) the minimal one. On the other hand, for k?1, T is said to be k-hyponormal if the operator matrix is positive. We prove that a 2-hyponormal operator T always satisfies the inequality T∗T∗,T]T?‖T‖2T∗,T], and as a result T is automatically weakly subnormal. Thus, a hyponormal operator T is 2-hyponormal if and only if there exists B such that BA∗=A∗T and is hyponormal, where A:=T∗,T]1/2. More generally, we prove that T is (k+1)-hyponormal if and and only if T is weakly subnormal and m.p.n.e.(T) is k-hyponormal. As an application, we obtain a matricial representation of the minimal normal extension of a subnormal operator as a block staircase matrix. |
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Keywords: | Weak subnormality k-hyponormality Minimal normal extensions of subnormal operators |
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