Functional Analysis and Its Applications最新文献

In a paper of Croot, Lev and Pach and a later paper of Ellenberg and Gijswijt, it was proved that for a group (G=G_0^n), where (G_0ne {1,-1}^m) is a fixed finite Abelian group and (n) is large, any subset (Asubset G) without 3-progressions (triples (x), (y), (z) of different elements with (xy=z^2)) contains at most (|G|^{1-c}) elements, where (c>0) is a constant depending only on (G_0). This is known to be false when (G) is, say, a large cyclic group. The aim of this note is to show that the algebraic property corresponding to this difference is the following: in the first case, a group algebra (mathbb{F}[G]) over a suitable field (mathbb{F}) contains a subspace (X) with codimension at most (|X|^{1-c}) such that (X^3=0). We discuss which bounds are obtained for finite Abelian (p)-groups and for some matrix (p)-groups: the Heisenberg group over (mathbb{F}_p) and the unitriangular group over (mathbb{F}_p). We also show how the method allows us to generalize the results of [14] and [12].

We associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse Kasteleyn matrix and hence all correlations. It may be seen as a quantization of the limit shape construction of Kenyon and the author. We also discuss various directions in which this correspondence may be generalized.

In this paper, we compare the Stone–Čech compactification (beta mathcal{P}(X)) of the space (mathcal{P}(X)) of Radon probability measures on a Tychonoff space (X), equipped with the weak topology, with the space (mathcal{P}(beta X)) of Radon probability measures on the Stone–Čech compactification (beta X) of the space (X). It is shown that for any noncompact metric space (X), the compactification (beta mathcal{P}(X)) does not coincide with (mathcal{P}(beta X)). We discuss the case of more general Tychonoff spaces and also the case of the Samuel compactification, for which the coincidence holds.

For a metric space (M), we prove existence of continuous maps ({M_n}^{infty}_{n=1}) associating to each compact set (K subset M), a probability measure (M_n(K)) with (operatorname{supp}(M_n(K)) = K) in such a way that the set ({M_n(K)}^{infty}_{n=1}) is dense in the space of probability measures on (K).

We answer a question posed by Vershik regarding connections between quasi-similarity of dynamical systems and Kolmogorov entropy. We prove that all Bernoulli actions of a given countably infinite group are quasi-similar to each other. The existence of non-Bernoulli actions in the same quasi-similarity class is an open problem. A notion opposite to quasi-similarity is that of disjointness (or independence) of actions. Pinsker proved that a deterministic action is independent from an action with completely positive entropy. Using joinings, we obtain the following generalization of Pinsker’s theorem: an action with zero (P)-entropy (an invariant defined by Kirillov and Kushnirenko) and an action with completely positive (P)-entropy are disjoint.

A simple convex approach and block techniques are used to obtain new sharpened versions of numerical radius inequalities for Hilbert space operators. These include comparisons of norms of operators, their Cartesian parts, their numerical radii, and the numerical radius of the product of two operators.

In this paper, we aim to prove an index theorem for linear relations and apply it to study the invertibility and the essential invertibility of certain upper triangular block relation matrices.

The article provides a concise overview of key concepts related to right quaternionic linear operators, quaternionic Hilbert spaces, and quaternionic Krein spaces. It then delves into the study of the quaternionic Krein space numerical range of a bounded right linear operator and the relationship between this numerical range and the (S)-spectrum of the operator. The article concludes by establishing spectral inclusion results based on the quaternionic Krein space numerical range and presenting the corresponding spectral inclusion theorems. In addition, we generalize some results to infinite dimensional quaternionic Krein spaces and give some examples.

An elliptic second-order differential operator (A_varepsilon=b(mathbf{D})^*g(mathbf{x}/varepsilon)b(mathbf{D})) on (L_2(mathbb{R}^d)) is considered, where (varepsilon >0), (g(mathbf{x})) is a positive definite and bounded matrix-valued function periodic with respect to some lattice, and (b(mathbf{D})) is a matrix first-order differential operator. Approximations for small (varepsilon) of the operator-functions (cos(tau A_varepsilon^{1/2})) and (A_varepsilon^{-1/2} sin (tau A_varepsilon^{1/2})) in various operator norms are obtained. The results can be applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation (partial^2_tau mathbf{u}_varepsilon(mathbf{x},tau) = - A_varepsilon mathbf{u}_varepsilon(mathbf{x},tau)).