泛函分析
Given a porous compact $K \subset \mathbb{R}^d$ and a continuity modulus $\omega$, we prove a quantitative Jackson-Bernstein type theorem on harmonic approximation. That is, a function $f$ belongs to the class $\mathrm{Lip}_{\omega}(K)$ if…
Sobolev type inequalities involving homogeneous elliptic canceling differential operators and rearrangement-invariant norms on the Euclidean space are considered. They are characterized via considerably simpler one-dimensional Hardy type…
We initiate a study of structural properties of the quotient algebra $\mathcal K(X)/\mathcal A(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the…
In [8] probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, have been used to show that for every infinite compact spaces K and L there exists a sequence $(\mu_n)$ of…
In analogy to a recent result by V. Fonf, M. Lin, and P. Wojtaszczyk, we prove the following characterizations of a Banach space $X$ with a basis. (i) $X$ is finite-dimensional if and only if every bounded, uniformly continuous, mean…
We give a self-contained and introductory account of some basic functional analytic tools needed to understand maximal monotone operators in Hilbert spaces. We review domains of (possibly unbounded) operators, closed sets and closed…
The main aim of this note is to prove a version of a celebrated theorem of Effros about transitive group actions in a non-metrizable setting, these parts have been formalized and verified with Lean by Lara Toledano. We do not claim any…
This book discusses the interactions between the (nonlinear) metric structure of Banach spaces and their linear asymptotic behavior. The overarching problem is to understand how the various linear structures of a Banach space are preserved…
We construct a class of Fourier multipliers whose associated operators are weak (1,1) bounded but fail to be weak (p, p) bounded for any 1 < p \leq \infty. Moreover, we show that this result is sharp.
In the present paper we improve Besov's recent result about upper estimates for the entropy numbers of Sobolev classes on a H\"{o}lder domain (in the case when the definition of the Sobolev class involves all partial derivatives of order…
In this paper, we establish the essential criteria for the hyponormality and quasinormality of the unbounded Toeplitz operator $T_{\varphi}$ with non-harmonic symbol, acting on the Fock-Sobolev space $F^{2, m}(\mathbb{C})$. The study shows…
We give a self-contained derivation of the Stinespring and Kraus structure theorems for completely positive maps using only scalar positive-definite kernels.
Answering questions raised in \cite{Leonetti, Uzcategui} we characterize ideals $\mathcal I\subseteq \mathcal P(\omega)$ such that $c_{0,\mathcal I}$ is complemented in $\ell_\infty$ as exactly those ideals for which the space $K_{\mathcal…
In this paper, we investigate the Schatten $p$-class ideals for $p >1$ as semi-inner product spaces in the sense of Giles and Lumer. Within this framework, we explore several geometric and analytic notions such as Birkhoff-James…
This paper investigates an iterative rank-one decomposition scheme for positive operators on a Hilbert space based on a residual-weighted congruence update. At each step the operator is compressed along a chosen unit vector while remaining…
In 1941, G. Gr\"unwald proved the convergence of a sequence of operators constructed using classical Lagrange interpolation at Chebyshev nodes. In this work, we establish a perturbed version of Gr\"unwald's result, thereby extending the…
We compare several versions of the quantitative Schur property of Banach spaces. We establish their equivalence up to multiplicative constants and provide examples clarifying when the change of constants is necessary. We also give exact…
For time-frequency localization operators, related to the short-time Fourier transform, with symbol $R\Omega$, we work out the exact large $R$ eigenvalue behavior for rotationally invariant $\Omega$ and conjecture that the same relation…
A recent discovery of Eldan and Gross states that there exists a universal $C>0$ such that for all Boolean functions $f:\{-1,1\}^n\to \{-1,1\}$, $$ \int_{\{-1,1\}^n}\sqrt{s_f(x)}d\mu(x) \ge C\text{Var}(f)\sqrt{\log…
The real anisotropic Littlewood's $4 / 3$ inequality is an extension of a famous result obtained in 1930 by J. E. Littlewood. It asserts that, for $a , b \in ( 0 , \infty )$, the following conditions are equivalent: $\bullet$ There is an…