Τ-measurable Operators Research Articles

Trace inequalities for measurable operators affiliated to a von Neumann algebra

Let φ be a trace on von Neumann algebra M, A, B ϵ M and ||B|| < 1, [A, B] = AB—BA. Then φ(|[A,B]|) ≤ 2 φ(|A|). Let τ be a faithful normal semifinite trace on M, S(M, τ) be the *-algebra of all τ-measurable operators. If A ϵ L2(M, τ) and Re A = λ|A| with λ ϵ {—1,1}, then A = λ|A|. An operator A ϵ L2(M, τ) is Hermitian if and only if τ(A2) = τ(A*A). Let positive operators A, B ϵ S(M, τ) be invertible in S(M, τ) and Y := (A-1 — B-1)(A — B). If Y, A1/2YA-1/2 ϵ L1(M, τ), then τ(Y) ≤ 0. Let an operator A ϵ S(M, τ) be hyponormal and A = B + iC be its Cartesian decomposition. If 1) BC ϵ L1(M, τ), оr 2) C = C3 ϵ M and [B, C] ϵ L1(M, τ), then A is normal.

New inequalities for (p,h)-convex functions for τ-measurable operators

The main goal of this article is to present new inequalities for (p, h)-convex and (p,h) log-convex functions for a non-negative super-multiplicative and super-additive function h. Our first main result will be h? (v/?) ? (h(1 ? v) f (a) + h(v) f (b))? ? f ? [((1 ? v)ap + vbp) 1 p] / (h(1 ? ?) f (a) + h(?) f (b))? ? f ? [((1 ? ?)ap + ?bp)1p] ? h? (1 ? v/1 ? ?), for the positive (p, h)-convex function f, when ? ? 1, p ? R\{0} and 0 ? v ? ? ? 1. This gives a generalization of an important result due to M. Sababheh [Linear Algebra Appl. 506 (2016), 588-602]. As applications of our results, we present many inequalities for the trace, and the symmetric norms for ?-measurable operators.

On 2×2 positive matrices of τ-measurable operators

On 2×2 positive matrices of τ-measurable operators

Strengthenings of Young-type inequalities and the arithmetic geometric mean inequality

Abstract In this paper, we present some generalizations and further refinements of Young-type inequality due to Choi [Math. Inequal. Appl. 21 (2018), 99–106], which strengthen the results obtained by Ighachane et al. [Math. Inequal. Appl. 23 (2020), 1079–1085]. As applications of these scalars results, we can get some inequalities for determinants, trace and p-norms of τ-measurable operators.

Some inequalities and majorization for products of τ-measurable operators

In this paper, for a semi-finite von Neumann algebra mathcal{M}, we study the Young, Hölder and Heinz means inequalities and extend results for τ-measurable operators. We obtain some refinements of the those inequalities for τ-measurable operators. We have also presented several inequalities in the sense of majorization.

On Young-type inequalities of measurable operators

Let be a semi-finite von Neumann algebra, be the set of all τ-measurable operators and be the generalized singular number of . We proved that if , and , then the Young-type inequality , for all t>0 holds.

A new generalization of refined young inequalities for τ-measurable operators

In this paper, by the arithmetic-geometric mean inequality, we give a new generalization of refined Young?s inequality. As applications we present some new generalizations of refinements of Young inequalities for the determinants, traces and p-norms of ?-measurable operators.

Trace and Commutators of Measurable Operators Affiliated to a Von Neumann Algebra

In this paper, we present new properties of the space L1(M, τ) of integrable (with respect to the trace τ ) operators affiliated to a semifinite von Neumann algebra M. For self-adjoint τ-measurable operators A and B, we find sufficient conditions of the τ -integrability of the operator λI −AB and the real-valuedness of the trace τ (λI − AB), where λ ∈ ℝ. Under these conditions, [A,B] = AB − BA ∈ L1(M, τ) and τ ([A,B]) = 0. For τ -measurable operators A and B = B2, we find conditions that are sufficient for the validity of the relation τ ([A,B]) = 0. For an isometry U ∈ M and a nonnegative τ -measurable operator A, we prove that U − A ∈ L1(M, τ) if and only if I − A, I − U ∈ L1(M, τ). For a τ -measurable operator A, we present estimates of the trace of the autocommutator [A∗,A]. Let self-adjoint τ -measurable operators X ≥ 0 and Y be such that [X1/2, YX1/2] ∈ L1(M, τ). Then τ ([X1/2, YX1/2]) = it, where t ∈ ℝ and t = 0 for XY ∈ L1(M, τ).

Submajorization inequalities for matrices of τ-measurable operators

Let be a semi-finite von Neumann algebra, be the set of all τ-measurable operators, be the generalized singular number of and be a concave function. We proved that if are normal operators in , then is submajorized by . As an application, we obtained that if x is a matrix of normal operators in , then is submajorized by .

Convergence in Measure and τ-Compactness of τ-Measurable Operators, Affiliated with a Semifinite von Neumann Algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.

Logarithmic submajorization, uniform majorization and Hölder type inequalities for [formula omitted]-measurable operators

We extend the notion of the determinant function Λ, originally introduced by T.Fack for τ-compact operators, to a natural algebra of τ-measurable operators affiliated with a semifinite von Neumann algebra which coincides with that defined by Haagerup and Schultz in the finite case and on which the determinant function is shown to be submultiplicative. Application is given to Hölder type inequalities via general Araki–Lieb–Thirring inequalities due to Kosaki and Han and to a Weyl-type theorem for uniform majorization.

Сходимость по мере и τ-компактность τ-измеримых операторов, ассоциированных с полуконечной алгеброй фон Неймана

Сходимость по мере и τ-компактность τ-измеримых операторов, ассоциированных с полуконечной алгеброй фон Неймана

On τ-essentially Invertibility of τ-measurable Operators

Let $$ \mathcal{M} $$ be a von Neumann algebra of operators on a Hilbert space and τ be a faithful normal semifinite trace on $$ \mathcal{M} $$ . Let I be the unit of the algebra $$ \mathcal{M} $$ . A τ-measurable operator A is said to be τ-essentially right (or left) invertible if there exists a τ-measurable operator B such that the operator I − AB (or I − BA) is τ-compact. A necessary and sufficient condition for an operator A to be τ-essentially left invertible is that A∗A (or, equivalently, $$ \sqrt{A^{\ast }A} $$ ) is τ-essentially invertible. We present a sufficient condition that a τ-measurable operator A not be τ-essentially left invertible. For τ-measurable operators A and P = P2 the following conditions are equivalent: 1. A is τ-essential right inverse for P; 2. A is τ-essential left inverse for P; 3. I − A,I − P are τ-compact; 4. PA is τ-essential left inverse for P. For τ-measurable operators A = A3, B = B3 the following conditions are equivalent: 1. B is τ-essential right inverse for A; 2. B is τ-essential left inverse for A. Pairs of faithful normal semifinite traces on $$ \mathcal{M} $$ are considered.

Generalizations of refined Young inequalities and reverse inequalities for τ-measurable operators

ABSTRACT In this paper, we present some generalizations of refinements of Young inequalities and reverse inequalities due to Furuichi, Choi as well as Manasrah and Kittaneh to cases of τ-measurable operators. Among other things, refined Young-type inequalities for operator traces, determinants and p-norms of positive τ-measurable operators are also provided.

The case of equality in Young's inequality for the s-numbers in semi-finite von Neumann algebras

For a semi-finite von Neumann algebra A, we study the case of equality in Young's inequality of s-numbers for a pair of τ-measurable operators a,b, and we prove that equality is only possible if |a|p=|b|q. We also extend the result to unbounded operators affiliated with A, and relate this problem with other symmetric norm Young inequalities.

Paranormal Measurable Operators Affiliated with a Semifinite von Neumann Algebra

Let M be a von Neumann algebra of operators on a Hilbert space H, τ be a faithful normal semifinite trace on M. We define two (closed in the topology of convergence in measure τ) classes P1 and P2 of τ-measurable operators and investigate their properties. The class P2 contains P1. If a τ-measurable operator T is hyponormal, then T lies in P1; if an operator T lies in Pk, then UTU* belongs to Pk for all isometries U from Mand k = 1, 2; if an operator T from P1 admits the bounded inverse T−1 then T−1 lies in P1. If a bounded operator T lies in P1 then T is normaloid, Tn belongs to P1 and a rearrangement μt(Tn) ≥ μt(T )n for all t > 0 and natural n. If a τ-measurable operator T is hyponormal and Tn is τ-compact operator for some natural number n then T is both normal and τ-compact. If an operator T lies in P1 then T 2 belongs to P1. If M= B(H) and τ = tr, then the class P1 coincides with the set of all paranormal operators onH. If a τ-measurable operator A is q-hyponormal (1 ≥ q > 0) and |A*| ≥ μ∞(A)I then Ais normal. In particular, every τ-compact q-hyponormal (or q-cohyponormal) operator is normal. Consider a τ-measurable nilpotent operator Z ≠ 0 and numbers a, b ∈ R. Then an operator Z*Z − ZZ* + aRZ + bSZ cannot be nonpositive or nonnegative. Hence a τ-measurable hyponormal operator Z ≠ 0 cannot be nilpotent.

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L1(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.

On some inequalities of τ-measurable operators

In this paper, we extended some inequalities which were proved By F. Kittaneh in [9] to the ?-measurable operators.

Kadec–Klee property for convergence in measure of noncommutative Orlicz spaces

This paper studies the Kadec–Klee property for convergence in measure of noncommutative Orlicz spaces Lφ(M˜,τ), where M˜ is the space of τ-measurable operators, and φ is an Orlicz function. We show that Lφ(M˜,τ) has the Kadec–Klee property in measure if and only if the φ satisfies the Δ2(∞) condition. As a corollary, the dual space and reflexivity of Lφ(M˜,τ) are given.

On weak* convergent sequences in duals of symmetric spaces of τ-measurable operators

It is shown that the pre-dual of a σ-finite von Neumann algebra has property (k) in the sense of Figiel, Johnson and Pelczynski [12]. This resolves in the affirmative an open question raised in [12]. It is shown further that a weakly sequentially complete symmetric space E of τ-measurable operators affiliated with a semifinite σ-finite von Neumann algebra has property (k).