Submajorization on $$\ell ^p(I)^+$$ determined by increasable doubly substochastic operators and its linear preservers

Ljubenović, Martin Z.; Rakić, Dragan S.

doi:10.1007/s43037-021-00143-9

Cited by 4 publications

(4 citation statements)

References 28 publications

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“…Let P σ be a doubly stochastic. For each E ∈ A with µ(E) < ∞, according to (9) in the proposition 1,…”

Section: Strictly Incoherent Operationmentioning

confidence: 99%

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Majorization and Semi-Doubly Stochastic Operators on $L^1(X)$

Manjegani¹,

Moein²

2021

Preprint

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This work devoted to extension of majorization based on semi doubly stochastic operators(denoted by SD(L 1 )) as a subclass of Markov operators but a larger class of doubly stochastic operator to L 1 (X) when X is a σ-finite measure space. We fully describe why such as these operators are more suitable for generalizing majorization relation. By using semi doubly stochastic operators we refuse Hiai's conjecture [6] and answer affirmatively to Mirsky's question [11] about extension of doubly stochastic matrix to infinite dimension as an infinite continuous counterpart of Hardy, Littlewood and Pólya's characterization of majorization. We collect a number of results of semi doubly stochastic operators such as equi-integrability of {Sf : S ∈ SD(L 1 )} when f is a fix element in L 1 (X) and consequently relatively weakly compact, and also strong relation of semi doubly stochastic operators and integral stochastic operators. Motivation of our work is application in quantum information theory like Nielson's Theorem, Uhlmann's Theorem and convertibility via incoherent operation. We explain extension of Uhlmann's Theorem to infinite dimension by means of semi doubly stochastic operators which leads to a quantum interpretation of majorization preserver. We introduce three types of maps which preserve majorization.

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“…Let P σ be a doubly stochastic. For each E ∈ A with µ(E) < ∞, according to (9) in the proposition 1,…”

Section: Strictly Incoherent Operationmentioning

confidence: 99%

“…Until recent decades, the main attention in majorization theory was paid to finite dimensional space, but recently because of its significant applications in a broad spectrum of fields, especially in quantum physics, considerable interest to infinite dimensional spaces appeared mathematically and physically [2,7,9,10,16].…”

Section: Introductionmentioning

confidence: 99%

Majorization and Semi-Doubly Stochastic Operators on $L^1(X)$

Manjegani¹,

Moein²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Until recent decades, the main attention in majorization theory was paid to finitedimensional spaces. However, recently, because of its significant applications in a broad spectrum of fields, especially in quantum physics, there arose a considerable mathematical and physical interest in infinite-dimensional spaces [4,10,12,13,19].…”

Section: Introductionmentioning

confidence: 99%

Majorization and semidoubly stochastic operators on $L^{1}(X)$

Manjegani

Moein

2023

J Inequal Appl

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This paper is devoted to a study of majorization based on semidoubly stochastic operators (denoted by $S\mathcal{D}(L^{1})$ S D ( L 1 ) ) on $L^{1}(X)$ L 1 ( X ) when X is a σ-finite measure space. We answer Mirsky’s question and characterize the majorization by means of semidoubly stochastic maps on $L^{1}(X)$ L 1 ( X ) . We prove some results on semidoubly stochastic operators such as a strong relation of semidoubly stochastic operators and integral stochastic operators and relatively weak compactness of $S_{f}=\{Sf : S\in S\mathcal{D}(L^{1})\}$ S f = { S f : S ∈ S D ( L 1 ) } for a fixed element $f\in L^{1}(X)$ f ∈ L 1 ( X ) by proving the equiintegrability of $S_{f}$ S f . We present a full characterization of majorization on a σ-finite measure space X.

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“…Particulary, if I is a finite set with n elements, we denote by Ω n and ω n for the set of all n × n doubly stochastic and doubly substochastic matrices, respectively. Definition 1.2 [3,11] Let A = [a ij ] be an I × I doubly substochastic matrix. Then A is called increasable if there exists an I × I doubly stochastic matrix D such that A ≤ D.…”

Section: Introductionmentioning

confidence: 99%

A generalization of a theorem of von Neumann

Eshkaftaki

2023

Preprint

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In 1953 von Neumann has proved that every n × n doubly substochastic matrix A can be increased to a doubly stochastic matrix, i.e., there is an n × n doubly stochastic matrix D for which A ≤ D. In this paper, we will discuss this result for a class of I × I doubly substochastic matrices. In fact, by a constructive method, we find an equivalent condition for existence of a doubly stochastic matrix D which satisfies A ≤ D, for all A ∈ A, where A is assumed a class of (finite or infinite) doubly substochastic matrices. Such a matrix D is called a cover of A. The uniqueness of the cover will also be discussed. Then we obtain an application of this concept to a system of (infinite) linear equations and inequalities. MSC(2020): 15B48, 15B51, 15A45, 47A50

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Submajorization on $$\ell ^p(I)^+$$ determined by increasable doubly substochastic operators and its linear preservers

Cited by 4 publications

References 28 publications

Majorization and Semi-Doubly Stochastic Operators on $L^1(X)$

Majorization and Semi-Doubly Stochastic Operators on $L^1(X)$

Majorization and semidoubly stochastic operators on $L^{1}(X)$

A generalization of a theorem of von Neumann

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