Search results for "Mathematics - Functional Analysis"
showing 10 items of 178 documents
Stochastic order characterization of uniform integrability and tightness
2013
We show that a family of random variables is uniformly integrable if and only if it is stochastically bounded in the increasing convex order by an integrable random variable. This result is complemented by proving analogous statements for the strong stochastic order and for power-integrable dominating random variables. Especially, we show that whenever a family of random variables is stochastically bounded by a p-integrable random variable for some p>1, there is no distinction between the strong order and the increasing convex order. These results also yield new characterizations of relative compactness in Wasserstein and Prohorov metrics.
Girsanov Theorem for Multifractional Brownian Processes
2017
In this article we will present a new perspective on the variable order fractional calculus, which allows for differentiation and integration to a variable order, i.e. one differentiates (or integrates) a function along the path of a regularity function. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the last 20 years. We develop a multifractional derivative operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional derivative operator, we are able to analyze a variety of new problems,…
Non linear pseudo-bosons versus hidden Hermiticity
2011
The increasingly popular concept of a hidden Hermiticity of operators (i.e., of their Hermiticity with respect to an {\it ad hoc} inner product in Hilbert space) is compared with the recently introduced notion of {\em non-linear pseudo-bosons}. The formal equivalence between these two notions is deduced under very general assumptions. Examples of their applicability in quantum mechanics are discussed.
Maximal Operators with Respect to the Numerical Range
2018
Let $\mathfrak{n}$ be a nonempty, proper, convex subset of $\mathbb{C}$. The $\mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $\mathfrak{n}$ and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the $\mathfrak{n}$-maximal operators are studied and some characterizations of these in terms of the resolvent set are given.
Quasiadditivity of Variational Capacity
2013
We study the quasiadditivity property (a version of superadditivity with a multiplicative constant) of variational capacity in metric spaces with respect to Whitney type covers. We characterize this property in terms of a Mazya type capacity condition, and also explore the close relation between quasiadditivity and Hardy's inequality.
Smooth surjections and surjective restrictions
2017
Given a surjective mapping $f : E \to F$ between Banach spaces, we investigate the existence of a subspace $G$ of $E$, with the same density character as $F$, such that the restriction of $f$ to $G$ remains surjective. We obtain a positive answer whenever $f$ is continuous and uniformly open. In the smooth case, we deduce a positive answer when $f$ is a $C^1$-smooth surjection whose set of critical values is countable. Finally we show that, when $f$ takes values in the Euclidean space $\mathbb R^n$, in order to obtain this result it is not sufficient to assume that the set of critical values of $f$ has zero-measure.
Clarkson-McCarthy inequalities with unitary and isometry orbits
2020
Abstract A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten p-classes for p > 2 is proved: if A , B are two n-by-n matrices, then there exists some pair of n-by-n unitary matrices U , V such that U | A + B 2 | p U ⁎ + V | A − B 2 | p V ⁎ ≤ | A | p + | B | p 2 . A similar statement holds for compact Hilbert space operators. Another improvement of McCarthy's inequality is given via the new operator parallelogramm law, | A + B | 2 ⊕ | A − B | 2 = U 0 ( | A | 2 + | B | 2 ) U 0 ⁎ + V 0 ( | A | 2 + | B | 2 ) V 0 ⁎ for some pair of 2n-by-n isometry matrices U 0 , V 0 .
Trace and density results on regular trees
2019
We give characterizations for the existence of traces for first order Sobolev spaces defined on regular trees.
Local maximal operators on fractional Sobolev spaces
2016
In this note we establish the boundedness properties of local maximal operators MG on the fractional Sobolev spaces Ws;p(G) whenever G is an open set in Rn, 0 < s < 1 and 1 < p < 1. As an application, we characterize the fractional (s;p)-Hardy inequality on a bounded open set by a Maz'ya-type testing condition localized to Whitney cubes. pq(G) whenever G is an open set in R n , 0 < s < 1 and 1 < p;q <1. Our main focus lies in the mapping properties of MG on a fractional Sobolev space W s;p (G) with 0 < s < 1 and 1 < p < 1, see Section 2 for the denition or (3) for a survey of this space. The intrinsically dened function space W s;p (G) on a given domain G coincides with the trace space F s …
Unbounded derivations and *-automorphisms groups of Banach quasi *-algebras
2018
This paper is devoted to the study of unbounded derivations on Banach quasi *-algebras with a particular emphasis to the case when they are infinitesimal generators of one parameter automorphisms groups. Both of them, derivations and automorphisms are considered in a weak sense; i.e., with the use of a certain families of bounded sesquilinear forms. Conditions for a weak *-derivation to be the generator of a *-automorphisms group are given.