Functional Analysis and Its Applications最新文献

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)).

For the Frobenius matrix accompanying an algebraic (differential) equation in a complex Banach algebra, the Cayley–Hamilton theorem is proved, which is used to obtain a representation of the resolvent.

Topological spaces with separately continuous Mal’tsev operation, called quasi-Mal’tsev spaces, are considered. The existence of the free quasi-Mal’tsev space generated by an arbitrary completely regular Hausdorff space is proved. It is shown that any quasi-Mal’tsev space is a quotient of a free quasi-Mal’tsev space. It is also shown that the topology of a free quasi-Mal’tsev space has a simple and natural description in terms of the generating space. Finally, it is proved that any completely regular Hausdorff quasi-Mal’tsev space is a retract of a quasi-topological group.

The full symmetric Toda system is a generalization of the open Toda chain, for which the Lax operator is a symmetric matrix of general form. This system is Liouville integrable and even superintegrable. Deift, Lee, Nando, and Tomei (DLNT) proposed the chopping method for constructing integrals of such a system. In the paper, a solution of Hamiltonian equations for the entire family of DLNT integrals is constructed by using the generalized QR factorization method. For this purpose, certain tensor operations on the space of Lax operators and special differential operators on the Lie algebra are introduced. Both tools can be interpreted in terms of the representation theory of the Lie algebra (mathfrak{sl}_n) and are expected to generalize to arbitrary real semisimple Lie algebras. As is known, the full Toda system can be interpreted in terms of a compact Lie group and a flag space. Hopefully, the results on the trajectories of this system obtained in the paper will be useful in studying the geometry of flag spaces.

In a recent paper, one of the authors proposed a construction of associative algebras which share a number of properties of the Yangians of series A but are more massive. We show that this construction admits a generalization which reveals a direct connection with a large family of double Poisson brackets on free associative algebras, which was described by Pichereau and Van de Weyer (in 2008).

We develop a differential-algebraic theory of the Mumford dynamical system. In the framework of this theory, we introduce the ((P,Q))-recursion, which defines a sequence of functions (P_1,P_2,ldots) given the first function (P_1) of this sequence and a sequence of parameters (h_1,h_2,dots). The general solution of the ((P,Q))-recursion is shown to give a solution for the parametric graded Korteweg–de Vries hierarchy. We prove that all solutions of the Mumford dynamical (g)-system are determined by the ((P,Q))-recursion under the condition (P_{g+1} = 0), which is equivalent to an ordinary nonlinear differential equation of order (2g) for the function (P_1). Reduction of the (g)-system of Mumford to the Buchstaber–Enolskii–Leykin dynamical system is described explicitly, and its explicit (2g)-parameter solution in hyperelliptic Klein functions is presented.

The asymptotic behavior of an exponential integral is studied in which the phase function has the form of a special deformation of the germ of a hyperbolic unimodal singularity of type (T_{4,4,4}). The integral under examination satisfies the heat equation, its Cole–Hopf transformation gives a solution of the vector Burgers equation in four-dimensional space-time, and its principal asymptotic approximations are expressed in terms of real solutions of systems of third-degree algebraic equations. The obtained analytical results make it possible to trace the bifurcations of an asymptotic structure depending on the parameter of the modulus of the singularity.

The paper is devoted to the study of the Kantorovich optimal transportation problem with nonlinear cost functional generated by a cost function depending on the conditional measures of the transport plan. The case of a cost function nonconvex in the second argument is considered. It is proved that this nonlinear Kantorovich problem with general cost function on a Souslin space can be reduced to the same problem with a convex cost function.

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.