Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II
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Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II Airat Bikchentaev1 Received: 2 September 2019 / Accepted: 17 February 2020 © Springer Nature Switzerland AG 2020
Abstract Let M be a von Neumann algebra of operators on a Hilbert space H and τ be a faithful normal semifinite trace on M. Let tτ be the measure topology on the ∗algebra S(M, τ ) of all τ -measurable operators. We define three tτ -closed classes P1 , P2 and P3 of τ -measurable operators and investigate their properties. The class P2 contains P1 ∪ P3 . If a τ -measurable operator T is hyponormal, then T lies in P1 ∩ P3 ; if an operator T lies in P3 , then U T U ∗ belongs to P3 for all isometries U from M. If a bounded operator T lies in P1 ∪ P3 then T is normaloid. If an operator T ∈ S(M, τ ) is p-hyponormal with 0 < p ≤ 1 then T ∈ P1 . If M = B(H) and τ = tr is the canonical trace, then the class P1 (resp., P3 ) coincides with the set of all paranormal (resp., ∗-paranormal) operators on H. Let A, B ∈ S(M, τ ) and A be p-hyponormal with 0 < p ≤ 1. If AB is τ -compact then A∗ B is τ -compact. Keywords Hilbert space · von Neumann algebra · Trace · Non-commutative integration · Measurable operator · Generalized singular value function · Paranormal operator · Hyponormal operator · Operator inequality Mathematics Subject Classification 46L10 · 47C15 · 46L51
1 Introduction It is well known that bounded hyponormal operators on a Hilbert space H have some interesting properties. For example, if A is a hyponormal operator then An ∞ = An∞ for every n ∈ N [20, Problem 162], here · ∞ denotes the uniform norm on B(H); every bounded hyponormal compact operator is normal [20, Problem 163].
This paper is dedicated to Professor P. G. Ovchinnikov.
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Airat Bikchentaev [email protected] Kazan Federal University, 18 Kremlyovskaya Str., Kazan, Russia 420008
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A. Bikchentaev
Fruitful generalizations of the notion of a hyponormal operator are the concepts of p-hyponormal [1], paranormal [17,23], and ∗-paranormal operators [3]. A number of modern authors study properties of such operators (see, for example, [29,30] and references in them). In this article, we obtain analogs of certain properties of bounded p-hyponormal, paranormal, and ∗-paranormal operators on H for some unbounded ones. Let M be a von Neumann operator algebra on a Hilbert space H, 1 be the unit of M, τ be a faithful normal semifinite trace on M, S(M, τ ) be the ∗-algebra of all τ -measurable operators, a number 0 < p < +∞ and L p (M, τ ) be the space of integrable (with respect to τ ) in pth degree operators. Let M1 = {X ∈ M : X ∞ = 1}, μ(·; X√) be the generalized singular value function of operator X ∈ S(M, τ ) and let |X | = X ∗ X . Assume that X ∞ = +∞ for all X ∈ S(M, τ )\M. In papers [6,8] we introduced two classes of τ -measurable operators P1 = {T ∈ S(M, τ ) : T 2 A∞ ≥ T A2∞ for all A ∈ M1 with T A ∈ M}, P2 = {T ∈ S(M, τ ) : μ(t; T 2 ) ≥ μ(t; T )2 for all t > 0} and investigated their properties. The classes P1 and P2 are c
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