arXiv subjects

Tomasz Szczepanski

Publications and source records attributed to Tomasz Szczepanski.

5 recordsLinked to original sources

arXiv2026

Rigidity of isometries on vector and normed lattices

We study rigidity phenomena for maps on vector and normed lattices arising from isometric, order-theoretic, and modulus-preserving conditions. We first investigate the role of strict convexity in Baker's nonsurjective version of the Mazur-Ulam theorem. In the process, we develop a theory of midpoint injective functions and investigate their connections with convexity and monotonicity. Using this framework, we prove a generalization of Baker's theorem. As our first main application, we show that every modulus isometry, i.e., a map $T:X\to Y$, where $X$ is a sublattice of a vector lattice $Y$, satisfying $$|T(x)-T(y)|=|x-y|, \qquad x,y\in X,$$ is affine and has a disjointness-preserving linear part. When $X$ is an ideal, the linear part is moreover a bijection of $X$ onto itself and an involution. In the normed lattice setting, we also show that every positive norm isometry into a strictly convex normed lattice is not only affine, but has a linear part that is a lattice homomorphism. Finally, we prove a nonlinear order-isometric rigidity theorem. After shifting to the origin, every order-preserving norm isometry between normed lattices whose codomain has a strictly monotone norm, preserves suprema and infima and is both disjointness preserving and disjointly additive. These conclusions do not force linearity, as shown by an explicit family of nonlinear isometries on $L_1[0,1]$.

math.FA↗

arXiv2026

On the chain of commuting operators on Banach spaces

An operator $T$ on a Banach space is said to be of chain $N$ if there exist non-scalar operators $S_1,\dots,S_{N-1}$ and a non-zero compact operator $K$ such that $$T \leftrightarrow S_1 \leftrightarrow S_2 \leftrightarrow \dots\leftrightarrow S_{N-1} \leftrightarrow K,$$ where $A\leftrightarrow B$ denotes $AB=BA$. We investigate this concept by identifying classes of operators that are of chain $N$ for some $N$. Our main result establishes that every weighted shift on $\ell_p$ ($1\leq p<\infty$) is of chain $3$, which in particular includes the class of non-Lomonosov operators studied by Hadwin et al. Furthermore, we provide an example of an operator on a separable Hilbert space that cannot be connected to a compact operator via a commuting chain of any length.

math.FA↗

arXiv2026

Nonlinear determination and phase retrieval under unimodular constraints

We study nonlinear determination problems in Hilbert spaces in which inner products are observed up to prescribed rotations in the complex plane. Given a Hilbert space $H$ and a subset $Θ$ of the unit circle $\mathbb{T}$, we say that a system $\mathbf{G}\subseteq H$ does $Θ$-phase retrieval ($Θ$-PR) if for all $f,h\in H$ the condition that for every $g\in\mathbf{G}$ there exists $θ_g\inΘ$ with $\langle f,g\rangle=θ_g\langle h,g\rangle$ forces $f=θh$ for some $θ\inΘ$. This framework unifies classical phase retrieval ($Θ=\mathbb{T}$) and sign retrieval ($Θ=\{1,-1\}$). For every countable $Θ$ we give a complete characterization of $Θ$-PR in terms of covers of $\mathbf{G}$ and geometric relations among vectors in the corresponding orthogonal complements, extending the complement-property characterization of Cahill, Casazza, and Daubechies. For cyclic phase sets we show that $Θ$-PR is equivalent to the existence of specific second-order recurrence relations. We apply this to obtain a sharp lattice density criterion for $Θ$-PR of exponential systems. For uncountable $Θ$ we obtain a topological dichotomy in the Fourier determination setting, showing that $Θ$-PR is characterized in terms of connectedness of $Θ$. We further develop a Möbius-invariant framework, proving that $Θ$-PR is preserved under circle automorphisms and is governed by projective invariants such as the cross ratio. Finally, in $\mathbb{C}^d$ we determine sharp impossibility thresholds and prove that for countable $Θ$ the property is generic once one passes the failure regime, yielding the minimal number of vectors required for $Θ$-PR.

math.FA↗

arXiv2025

Minimal dispersion on the sphere

The minimal spherical cap dispersion ${\rm disp}_{\mathcal{C}}(n,d)$ is the largest number $\varepsilon\in (0,1]$ such that, for every $n$ points on the $d$-dimensional Euclidean unit sphere $\mathbb{S}^d$, there exists a spherical cap with normalized area $\varepsilon$ not containing any of these points. We study the behavior of ${\rm disp}_{\mathcal{C}}(n,d)$ as $n$ and $d$ grow to infinity. We develop connections to the problems of sphere covering and approximation of the Euclidean unit ball by inscribed polytopes. Existing and new results are presented in a unified way. Upper bounds on ${\rm disp}_{\mathcal{C}}(n,d)$ result from choosing the points independently and uniformly at random and possibly adding some well-separated points to close large gaps. Moreover, we study dispersion with respect to intersections of caps.

math.MG↗

arXiv2015

Superpoissonian shot noise in organic magnetic tunnel junctions

Organic molecules have recently revolutionized ways to create new spintronic devices. Despite intense studies, the statistics of tunneling electrons through organic barriers remains unclear. Here we investigate conductance and shot noise in magnetic tunnel junctions with PTCDA barriers a few nm thick. For junctions in the electron tunneling regime, with magnetoresistance ratios between 10 and 40\%, we observe superpoissonian shot noise. The Fano factor exceeds in 1.5-2 times the maximum values reported for magnetic tunnel junctions with inorganic barriers, indicating spin dependent bunching in tunneling. We explain our main findings in terms of a model which includes tunneling through a two level (or multilevel) system, originated from interfacial bonds of the PTCDA molecules. Our results suggest that interfaces play an important role in the control of shot noise when electrons tunnel through organic barriers.

cond-mat.mes-hall↗