Young’s (in)equality for compact operators

Larotonda, Gabriel

doi:10.4064/sm8427-5-2016

Cited by 3 publications

(3 citation statements)

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“…Yet another different proof is based on the s-numbers of operators and majorization theory, i.e. the proof given by M. Manjegani in [Ma07]; that proof depends on the solution of the case of equality in Young's inequality for nuclear operators which was given in [AF03] (for the case of equality for the singular values of compact operators, see [La16]).…”

Section: Introductionmentioning

confidence: 99%

The case of equality in Hölder's inequality for matrices and operators

Larotonda¹

2018

Mathematical Proceedings of the Royal Irish Academy

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Let p > 1 and 1/p + 1/q = 1. Consider Hölder's inequality ab * 1 ≤ a p b q for the p-norms of some trace (a, b are matrices, compact operators, elements of a finite C * -algebra or a semi-finite von Neumann algebra). This note contains a simple proof (based on the case p = 2) of the fact that equality holds iff |a| p = λ|b| q for some λ ≥ 0.Since the main result of this note is based on it, let us start by recalling the well-known Cauchy-Schwarz inequality with precision; for a proof see Proposition 2.1.3 in Kadison and Ringrose's book [KR83]. Lemma 1.1. Let x, y ∈ A and set x, y = τ (xy * ), · 2 = x, x . Then | x, y | ≤ x 2 y 2 .Moreover, if τ (xy * ) = x 2 y 2 , then x = λy for some λ ≥ 0.Remark 1.2. Let a = u|a| be the polar decomposition of a ∈ A, then u * u|a| = |a|. Write a = xy * with x = u|a| 1/2 and y = |a| 1/2 , and use Cauchy-Schwarz to obtain |τ (a)| ≤ τ (|a|) with (finite) equality τ (a) = τ |a| = 1 if and only if a = |a| ≥ 0.

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Section: Introductionmentioning

confidence: 99%

The case of equality in Hölder's inequality for matrices and operators

Larotonda¹

2018

Mathematical Proceedings of the Royal Irish Academy

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“…In this paper, we establish an analogue for the case of equality in the setting of operators affiliated to semi-finite von Neumann algebras. For more references and further discussion on the subject of Young's inequality for matrices and operators, we refer the reader to [12] where the proof is given for the particular case of compact operators in B(H) -the discrete (or atomic measure) case-of this fact. In particular, we remark that it was the fundamental paper by T. Ando [1] which initiated the study of Young's inequality for the singular values of n × n matrices.…”

Section: Introductionmentioning

confidence: 99%

“…The emphasis in this paper is in the measure theoretic approach to operators affiliated with a semi-finite von Neumann algebra, since the approach by induction used in [12] is not at hand. The inequality for s-numbers of operators a, b affiliated with a semi-finite von Neuman algebra A, is stated as µ s (ab * ) ≤ µ s ( 1 /p |a| p + 1 /q |b| q ) , s > 0 (1) and extended here to unbounded operators; we are interested in the case of equality.…”

Section: Introductionmentioning

confidence: 99%