Functional Analysis and Its Applications最新文献

In his paper “The Mumford dynamical system and hyperelliptic Kleinian functions” [Funkts. Anal. Prilozhen. 57 (4), 27–45 (2023)] Victor Buchstaber developed the differential-algebraic theory of the Mumford dynamical system. The key object of this theory is the ((P,Q))-recursion introduced in his paper.

In the present paper, we further develop the theory of the ((P,Q))-recursion and describe its connections to the Korteweg–de Vries hierarchy, the Lenard operator, and the Gelfand–Dikii recursion.

We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, this is an invariant of this function with respect to a certain group of transformations of variables; on the other hand, this is a special probability measure in the space of matrices (tensors) that is invariant under actions of natural infinite permutation groups. The intricate interplay of both interpretations of matrix (tensor) distributions makes them an important subject of modern functional analysis. We formulate and prove a theorem that, under certain conditions on a measurable function of two variables, its matrix distribution is a complete invariant.

It is proved that the generic extensions of a dynamical system inherit the triviality of pairwise independent self-joinings. This property is related to well-known problems of joining theory and to Rokhlin’s famous multiple mixing problem.

The multiplicative version of the Gleason–Kahane–Żelazko theorem for (C^*)-algebras given by Brits et al. in [4] is extended to maps from (C^*)-algebras to commutative semisimple Banach algebras. In particular, it is proved that if a multiplicative map (phi) from a (C^*)-algebra (mathcal{U}) to a commutative semisimple Banach algebra (mathcal{V}) is continuous on the set of all noninvertible elements of (mathcal{U}) and (sigma(phi(a)) subseteq sigma(a)) for any (a in mathcal{U}), then (phi) is a linear map. The multiplicative variation of the Kowalski–Słodkowski theorem given by Touré et al. in [14] is also generalized. Specifically, if (phi) is a continuous map from a (C^*)-algebra (mathcal{U}) to a commutative semisimple Banach algebra (mathcal{V}) satisfying the conditions (phi(1_mathcal{U})=1_mathcal{V}) and (sigma(phi(x)phi(y)) subseteq sigma(xy)) for all (x,y in mathcal{U}), then (phi) generates a linear multiplicative map (gamma_phi) on (mathcal{U}) which coincides with (phi) on the principal component of the invertible group of (mathcal{U}). If (mathcal{U}) is a Banach algebra such that each element of (mathcal{U}) has totally disconnected spectrum, then the map (phi) itself is linear and multiplicative on (mathcal{U}). It is shown that a similar statement is valid for a map with semisimple domain under a stricter spectral condition. Examples which demonstrate that some hypothesis in the results cannot be discarded.

We study the problem of describing the triples ((Omega,g,mu)), (mu=rho,dx), where (g= (g^{ij}(x))) is the (co)metric associated with a symmetric second-order differential operator (mathbf{L}(f) = frac{1}{rho}sum_{ij} partial_i (g^{ij} rho,partial_j f)) defined on a domain (Omega) of (mathbb{R}^d) and such that there exists an orthonormal basis of (mathcal{L}^2(mu)) consisting of polynomials which are eigenvectors of (mathbf{L}) and this basis is compatible with the filtration of the space of polynomials by some weighted degree.

In a joint paper of D. Bakry, M. Zani, and the author this problem was solved in dimension 2 for the usual degree. In the present paper we solve it still in dimension 2 but for a weighted degree with arbitrary positive weights.

We consider partial theta series associated with periodic sequences of coefficients, namely, (Theta(tau):= sum_{n>0} n^nu f(n) e^{ipi n^2tau/M}), where (nuinmathbb{Z}_{ge0}) and

(fcolonmathbb{Z} to mathbb{C}) is an (M)-periodic function. Such a function (Theta) is analytic in the half-plane ({ operatorname {Im}tau>0}) and in the asymptotics of (Theta(tau)) as (tau) tends nontangentially to any (alphainmathbb{Q}) a formal power series appears, which depends on the parity of (nu) and (f). We discuss the summability and resurgence properties of these series; namely, we present explicit formulas for their formal Borel transforms and their consequences for the modularity properties of (Theta), or its “quantum modularity” properties in the sense of Zagier’s recent theory. The discrete Fourier transform of (f) plays an unexpected role and leads to a number-theoretic analogue of Écalle’s “bridge equations.” The main thesis is: (quantum) modularity (=) Stokes phenomenon (+) discrete Fourier transform.

We construct a Springer-type resolution of singularities of the odd nilpotent cone of the orthosymplectic Lie superalgebras (mathfrak{osp}(m|2n)).

We give an estimate of the rate of convergence to zero of the norms of remotest projections on three subspaces of a Hilbert space with zero intersection for starting vectors in the sum of orthogonal complements to these subspaces.

The notion of the limit spectral measure of a metric triple (i.e., a metric measure space) is defined. If the metric is square integrable, then the limit spectral measure is deterministic and coincides with the spectrum of the integral operator on (L^2(mu)) with kernel (rho). An example in which there is no deterministic spectral measure is constructed.

New results on the strong solvability in Sobolev spaces of the quasilinear Venttsel’ problem for parabolic equations with discontinuous leading coefficients are obtained.