Boundary triplets and generalized resolvents of isometric operators in a Pontryagin space

The notions of the boundary triplet of an isometric operator V in the Pontryagin space and the corresponding function Weyl are introduced. Proper extensions of the isometric operator V, their spectra, and canonical and generalized resolvents of the operator V are described.     

On generalized resolvents of Hermitian operators in Krein spaces

Various classes of extensions and generalized resolvents of Hermitian operators acting in Krein spaces are described in terms of abstract boundary conditions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1134–1147, September, 1994.     

Analytically invariant spectral resolvents of closed operators

The concept of analytically invariant subspace, in the framework of spectral resolvents, is extended to unbounded closed operators. One obtains necessary and sufficient conditions for a spectral resolvent to be analytically invariant. An application, concerning a strong type of spectral decomposition for closed operators with the spectra on the real line, concludes this study.     

Extensions and generalized resolvents of closed dissipative operators

Let B be a closed dissipative operator in a Hilbert space ? with an arbitrary domain of definition. We investigate briefly the problem of describing all the closed (and, in particular, the closed maximal) dissipative extensions B of the operator B. Following this we introduce the concept of a generalized resolvent of a closed dissipative operator with an arbitrary domain of definition, and we study the fundamental properties of generalized resolvents.     

Description of generalized resolvents and characteristic matrices of differential operators in terms of the boundary parameter

Mathematical Notes - We supplement and further develop well-known results due to Shtraus on the generalized resolvents and spectral functions of the minimal operator L 0 generated by a formally...     

Asymptotic properties of the spectrum of a differential operator in a space of vector-valued functions

An investigation of the asymptotic properties of a two-dimensional Sturm-Liouville system with the rotation of vectors taken into consideration.Translated from Matematicheskie Zametki, Vol. 9, No. 6, pp. 667–676, June, 1971.     

Infinite order differential operators in spaces of entire functions

Differential operators ?(Δ_θ,ω), where ? is an exponential type entire function of a single complex variable and Δ_θ,ω=(θ+ωz)D+zD², D=∂/∂z, , θ?0, , acting in the spaces of exponential type entire function are studied. It is shown that, for ω?0, such operators preserve the set of Laguerre entire functions provided the function ? also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane . The operator exp(aΔ_θ,ω), a?0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all . An integral representation of exp(aΔ_θ,ω), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δ_θ,ω.     

On a class of partial differential equations of even order

Summary The general solution for a class of equations of even order is expressed as a sum of solutions of equations of second order. To Mauro Picone on his 70^th birthday. This research was supported in part by the United States Air Force under Contract N^o. AF(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

Spectral properties of the family of even order differential operators with a summable potential

A boundary value problem for a higher order differential operator with separated boundary conditions is considered. The asymptotics of solutions of the corresponding differential equation for large values of the spectral parameter is studied. The indicator diagram of the equation for the eigenvalues is studied. The asymptotic behavior of eigenvalues and the formula for calculation of eigenfunctions of the studied operator is obtained in different sectors of the indicator diagram.     

Groups of partial differential operators and the generalized Bessel functions

Considering the generalized Bessel functions satisfying a special ordinary differential equation of mth order, we derive some addition theorems and generating functions with the help of some algebra constructed for a group of first order partial differential operators grounding on the recurrence relations for these functions.     

Absolutely summing operators and integration of vector-valued functions

Let (Ω,Σ,μ) be a complete probability space and an absolutely summing operator between Banach spaces. We prove that for each Dunford integrable (i.e., scalarly integrable) function the composition u○f is scalarly equivalent to a Bochner integrable function. Such a composition is shown to be Bochner integrable in several cases, for instance, when f is properly measurable, Birkhoff integrable or McShane integrable, as well as when X is a subspace of an Asplund generated space or a subspace of a weakly Lindelöf space of the form C(K). We also study the continuity of the composition operator f?u○f. Some other applications are given.     

On periodic solutions for even order differential equations

A class of sufficient conditions are obtained for the existence of a unique 2π$2\pi$-periodic solution of even order differential equations, employing an initial value problem.