Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis

For the equationL ₀ x(t)+L ₁x(t)+...+L _n x ⁽ⁿ⁾(t)=O, whereL _k,k=0,1,...,n, are operators acting in a Banach space, we establish criteria for an arbitrary solutionx(t) to be zero provided that the following conditions are satisfied:x ^(1–1) (a)=0, 1=1, ..., p, andx ^(1–1) (b)=0, 1=1,...,q, for - <a< b< (the case of a finite segment) orx ^(1–1) (a)=0, 1=1,...,p, under the assumption that a solutionx(t) is summable on the semiaxista with its firstn derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 279–292, March, 1994.This research was supported by the Ukrainian State Committee on Science and Technology.     

Completeness of root vectors of a Keldysh pencil perturbed by an analytic operator-valued function S(λ) with S(∞) = 0

The multiple completeness of the root vectors of the pencil $$L(\lambda ) = I - T_0 - \lambda T_1 H - \ldots - \lambda ^{n - 1} T_{n - 1} H^{n - 1} - \lambda ^n H^n - S(\lambda ),$$ where I is the identity operator in the separable Hilbert space ,S (λ) is an operator-valued function analytic for ¦λ¦ > η with S(∞) = 0, and T_k and H are completely continuous operators, is studied. The method suggested in this note for proving the completeness does not use the factorization theorems, due to which we can remove certain restrictions on the function S(λ) connected with the application of the factorization theorems.     

On One Extremal Problem for a Seminorm on the Space l<Subscript>1</Subscript> with Weight

Let be a nondecreasing sequence of positive numbers and let l _1,α be the space of real sequences for which . We associate every sequence ξ from l _1,α with a sequence , where ϕ(·) is a permutation of the natural series such that , j ∈ ℕ. If p is a bounded seminorm on l _1,α and , then

Uniqueness of Solutions of Some Nonlocal Boundary-Value Problems for Operator-Differential Equations on a Finite Segment

For the equation L ₀ x(t) + L ₁ x ⁽¹⁾(t) + ... + L _n x ⁽ⁿ⁾(t) = 0, where L _k, k = 0, 1, ... , n, are operators acting in a Banach space, we formulate conditions under which a solution x(t) that satisfies some nonlocal homogeneous boundary conditions is equal to zero.     

Equivalence in Ln[0, 1] of the system ei2πkx (k=0, ±1, ...) and the system of the eigenfunctions of an ordinary functional-differential operator

Translated from Matematicheskie Zametki, Vol. 49, No. 1, pp. 47–55, January, 1991.     

Equivalence of the derived chains corresponding to a boundary-value problem on a finite interval,for polynomial operator pencils

Under investigation is the equivalence of derived chains constructed from root vectors of polynomial pencils of operators acting in a Hilbert space. These derived chains correspond to various boundary—value problems on a finite interval for an operator—differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 83–95, January, 1990.     

Basis and completeness theorems for the characteristic vectors of hyperbolic operator-functions

Siberian Mathematical Journal -     

On an upper bound for the number of characteristic values of an operator function

We prove a theorem on an upper bound for the number of characteristic values of an operator-valued function that is holomorphic and bounded in a domain. This estimate is similar to the well-known inequality for zeros of a number function that is holomorphic and bounded in a domain. We derive several corollaries of the theorem proved, in particular, a statement on an estimate of the number of characteristic values of polynomial bundles of operators that lie in a given disk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 211–224, February, 1998. This work was financially supported by the Foundation for Fundamental Research of the Ukrainian Ministry of Science.