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Register a new userLineability, spaceability, and additivity cardinals for Darboux-like functions
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We introduce the concept of {em maximal lineability cardinal number}, $mL(M)$, of a subset $M$ of a topological vector space and study its relation to the cardinal numbers known as: additivity $A(M)$, homogeneous lineability $HL(M)$, and lineability $LL(M)$ of $M$. In particular, we will describe, in terms of $LL$, the lineability and spaceability of the families of the following Darboux-like functions on $real^n$, $nge 1$: extendable, Jones, and almost continuous functions.
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Sub-additive and super-additive inequalities for concave and convex functions have been generalized to the case of matrices by several authors over a period of time. These lead to some interesting inequalities for matrices, which in some cases coincide with, and in other cases are at variance with the corresponding inequalities for real numbers. We survey some of these matrix inequalities and do further investigations into these. We introduce the novel notion of dominated majorization between the spectra of two Hermitian matrices $B$ and $C$, dominated by a third Hermitian matrix $A$. Based on an explicit formula for the gradient of the sum of the $k$ largest eigenvalues of a Hermitian matrix, we show that under certain conditions dominated majorization reduces to a linear majorization-like relation between the diagonal elements of $B$ and $C$ in a certain basis. We use this notion as a tool to give new, elementary proofs for the sub-additivity inequality for non-negative concave functions first proved by Bourin and Uchiyama and the corresponding super-additivity inequality for non-negative convex functions first proven by Kosem. Finally, we present counterexamples to some conjectures that Andos inequality for operator convex functions could more generally hold, e.g. for ordinary convex, non-negative functions.
In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not a.e.~pointwise, uniformly but not pointwise convergent, and uniformly convergent but not in $L^1$-norm, are analyzed. These findings extend and complement a number of earlier results by several authors.
It is proved the existence of large algebraic structures break --including large vector subspaces or infinitely generated free algebras-- inside, among others, the family of Lebesgue measurable functions that are surjective in a strong sense, the family of nonconstant differentiable real functions vanishing on dense sets, and the family of non-continuous separately continuous real functions. Lineability in special spaces of sequences is also investigated. Some of our findings complete or extend a number of results by several authors.
Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p in [1,infty]$) $Z(ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(ell_infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X setminus Z(X)$ and its algebraic genericity.
For positive semidefinite matrices $A$ and $B$, Ando and Zhan proved the inequalities $||| f(A)+f(B) ||| ge ||| f(A+B) |||$ and $||| g(A)+g(B) ||| le ||| g(A+B) |||$, for any unitarily invariant norm, and for any non-negative operator monotone $f$ on
$[0,infty)$ with inverse function $g$. These inequalities have very recently been generalised to non-negative concave functions $f$ and non-negative convex functions $g$, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities $||| f(A)-f(B) ||| le ||| f(|A-B|) |||$, and $||| g(A)-g(B) ||| ge ||| g(|A-B|) |||$, obtained by Ando, for operator monotone $f$ with inverse $g$, also have a similar generalisation to non-negative concave $f$ and convex $g$. We answer exactly this question, in the negative for general matrices, and affirmatively in the special case when $Age ||B||$. In the course of this work, we introduce the novel notion of $Y$-dominated majorisation between the spectra of two Hermitian matrices, where $Y$ is itself a Hermitian matrix, and prove a certain property of this relation that allows to strengthen the results of Bourin-Uchiyama and Kosem, mentioned above.
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