Related papers: Necessary and sufficient condition for compactness…
We give necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss structure of such operators in the case of self-adjoint and normal operators separately.…
We study extension operators on Sobolev spaces with decreasing integrability on the base of set functions associated with the operator norms. Sharp necessary conditions in the terms of the generalized density condition and the terms of weak…
We show that a compact operator $A$ is a multiple of a positive semi-definite operator if and only if $$ \sigma(AB) \subseteq \overline{W(A)W(B)}, \quad\text{for all (rank one) operators $B$}. $$ An example of a normal operator is given to…
We give new necessary and sufficient conditions for higher order convex ordering. These results generalize the Levin-Ste\v{c}kin theorem (1960) on convex ordering. The obtained results can be useful in the study of the Hermite-Hadamard type…
The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency…
We prove sufficient and necessary conditions for compactness of Calder\'on-Zygmund operators on the endpoint from $L^{\infty }(\mathbb R)$ into ${\rm CMO}(\mathbb R)$. We use this result to prove compactness on $L^{p}(\mathbb R)$ with…
The necessary and sufficient conditions for differentiability of a function of several real variables stated and proved and its ramifications discussed.
This paper presents two general criteria to determine spaceability results in the complements of unions of subspaces. The first criterion applies to countable unions of subspaces under specific conditions and is closely related to the…
The main goal of this note is to characterize the necessary and sufficient conditions for a composition operator to act between spaces of mappings of bounded Wiener variation in a normed-valued setting. The necessary and sufficient…
We establish necessary and sufficient conditions for the boundedness of the relativistic Schr\"odinger operator $\mathcal{H} = \sqrt{-\Delta} + Q$ from the Sobolev space $W^{1/2}_2 (\R^n)$ to its dual $W^{-1/2}_2 (\R^n)$, for an arbitrary…
Two necessary and sufficient conditions for an operator to be semi-normal are revealed. For a Volterra integration operator the set where the operator and its adjoint are metrically equal is described.
In this paper we solve a long standing problem about the bilinear $T1$ theorem to characterize the (weighted) compactness of bilinear Calder\'{o}n-Zygmund operators. Let $T$ be a bilinear operator associated with a standard bilinear…
We give sufficient conditions for the essential self-adjointness of perturbed biharmonic operators acting on sections of a Hermitian vector bundle over a Riemannian manifold with additional assumptions, such as lower semi-bounded Ricci…
This is the second in a series of articles aimed at exploring the relationship between the complexity classes of P and NP. The research in this article aims to find conditions of an algorithmic nature that are necessary and sufficient to…
Motivated by recent work on ordinal embedding (Kleindessner and von Luxburg, 2014), we derive large sample consistency results and rates of convergence for the problem of embedding points based on triple or quadruple distance comparisons.…
The boundedness and compactness of weighted composition operators on the Hardy space ${\mathcal H}^2$ of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class…
We derive a precise general relation between the entropy of a compact operator and its eigenvalues. It is then shown how this result along with the underlying philosophy can be applied to improve substantially on the best known…
Let $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be a polynomial and $\mathcal{Z}(f)$ its zero set. In this paper, in terms of the so-called Newton polyhedron of $f,$ we present a necessary criterion and a sufficient condition for the…
We investigate an extension of Schauder's theorem by studying the relationship between various $s$-numbers of an operator $T$ and its adjoint $T^*$. We have three main results. First, we present a new proof that the approximation number of…
We give a necessary and sufficient condition for an n-hypercontraction to be similar to the backward shift operator in a weighted Bergman space. This characterization serves as a generalization of the description given in the Hardy space…