Discontinuous Sturm-Liouville problems involving an abstract linear operator

In this paper we introduce to consideration a new type boundary value problems consisting of an ``Sturm-Liouville" equation on two disjoint intervals as $$-p(x)y^{\prime \prime }+ q(x)y+\mathfrak{B}y|_{x} = \mu y , x\in [a,c)\cup(c,b]$$ together with two end-point conditions whose coefficients depend linearly on the eigenvalue parameter, and two supplementary so-called transmission conditions, involving linearly left-hand and right-hand values of the solution and its derivatives at point of interaction $x=c,$ where $\mathfrak{B}:L_{2}(a,c)\oplus L_{2}(c,b)\rightarrow L_{2}(a,c)\oplus L_{2}(c,b)$ is an abstract linear operator, non-selfadjoint in general. For self-adjoint realization of the pure differential part of the main problem we define ``alternative" inner products in Sobolev spaces, ``incorporating" with the boundary-transmission conditions. Then by suggesting an own approaches we establish such properties as topological isomorphism and coercive solvability of the corresponding nonhomogeneous problem and prove compactness of the resolvent operator in these Sobolev spaces. Finally, we prove that the spectrum of the considered eigenvalue problem is discrete and derive asymptotic formulas for the eigenvalues. Note that the obtained results are new even in the case when the equation is not involved an abstract linear operator $\mathfrak{B}.$     

Descriptive properties of density preserving autohomeomorphisms of the unit interval

We prove that density preserving homeomorphisms form a Π ₁ ¹ -complete subset in the Polish space ? of all increasing autohomeomorphisms of unit interval.     

Existence results for operator equations in abstract spaces and an application

In this paper we apply the recursion method presented in (Heikkilä and Lakshmikantham, 1994) to derive fixed point theorems and existence results for operator equations in partially ordered sets. The obtained results, combined with related results of (Heikkilä, 2002) are then applied to operator equations in ordered Hilbert spaces and to a functional elliptic boundary value problem.     

Mapping properties of an integral operator

In this paper, we consider certain subclasses of analytic functions with bounded radius and bounded boundary rotation and study the mapping properties of these classes under an integral operator.     

Spectral properties of an even-order differential operator

We present the spectral properties of an even-order differential operator whose domain is described by periodic and antiperiodic boundary conditions or the Dirichlet conditions. We derive an asymptotic formula for the eigenvalues, estimates for the deviations of spectral projections, and estimates for the equiconvergence rate of spectral decompositions. Our asymptotic formulas for eigenvalues refine well-known ones.     

Applications of operator spaces to abstract harmonic analysis

We give a survey of how the relatively young theory of operator spaces has led to a deeper understanding of the Fourier algebra of a locally compact group (and of related algebras).     

On the evolution operator for abstract parabolic equations

We find a new construction of the evolution operatorG(t, s) associated to a family {A(t), 0≦t≦T} of generators of analytic semigroups in a Banach spaceX. We study the dependence ofG (t, s) ont ands, and we give regularity results for the solution of the i.v.p.u′(t)=A(t)u(t)+f(t),u(0)=x.     

On the spectral properties of an operator pencil

For a broad class of iterative algorithms for solving saddle point problems, the study of the convergence and of the optimal properties can be reduced to an analysis of the eigenvalues of operator pencils of a special form. A new approach to analyzing spectral properties of pencils of this kind is proposed that makes it possible to obtain sharp estimates for the convergence rate.     

On abstract and almost-abstract density topologies

We concentrate on topologies generated by lower and almost-lower density operators on measurable spaces. Among others the existence of the smallest in the sense of inclusion abstract density topology on measurable space is investigated. Moreover, the separation axioms for such topologies are studied.     

Categorical abstract algebraic logic: The categorical Suszko operator

Czelakowski introduced the Suszko operator as a basis for the development of a hierarchy of non‐protoalgebraic logics, paralleling the well‐known abstract algebraic hierarchy of protoalgebraic logics based on the Leibniz operator of Blok and Pigozzi. The scope of the theory of the Leibniz operator was recently extended to cover the case of, the so‐called, protoalgebraic π‐institutions. In the present work, following the lead of Czelakowski, an attempt is made at lifting parts of the theory of the Suszko operator to the π‐institution framework. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Green's function and invariant density for an integro-differential operator of second order

Summary We construct the Green function for a parabolic- elliptic integro- differential operator of second order with boundary conditions given by a first order operator, both with Hölder continuous coefficients. Also, we prove the existence of a unique invariant density associated with the corresponding reflected diffusion process with jumps. This allows us to study the asymptotic behaviour of the solution for some oblique derivative problems.This research have been supported in part by Ministero P.I. « Progetto Caleolo delle Variazioni » and N.S.F. under grant No. DMS-8702236.     

On density of transitive operator algebras

In this paper we study the transitive algebra question by considering the invariant subspace problem relative to von Neumann algebras. We prove that the algebra (not necessarily ∗) generated by a pair of sums of two unitary generators of L(F_∞) and its commutant is strong-operator dense in B(H). The relations between the transitive algebra question and the invariant subspace problem relative to some von Neumann algebras are discussed.     

Similarity of an operator to an isometric operator

Leningrad State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 2, pp. 77–78, April–June, 1989.