Related papers: Some identification problems for integro-different…
A Hilbert space operator $U$ is called universal (in the sense of Rota) if every Hilbert space operator is similar to a multiple of $U$ restricted to one of its invariant subspaces. It follows that the Invariant Subspace Problem for Hilbert…
A parabolic partial differential equation $u'_t(t,x)=Lu(t,x)$ is considered, where $L$ is a linear second-order differential operator with time-independent coefficients, which may depend on $x$. We assume that the spatial coordinate $x$…
This paper is concerned with the solvability of some abstract differential equation of type $$\dot u(t) + Au(t) + Bu(t) \ni f(t), t \in (0,T], u(0) = 0,$$ where $A$ is a linear selfadjoint operator and $B$ is a nonlinear(possibly…
We study the asymptotic behaviour of eigenvalues and eigenfunctions of a boundary value problem for the Sturm-Liouville operator with general boundary conditions and the weight function perturbed by the so-called $\delta'$-like sequence…
We consider the Sobolev embedding operator $E_s : H^s(\Omega) \to L_2(\Omega)$ and its role in the solution of inverse problems. In particular, we collect various properties and investigate different characterizations of its adjoint…
Integral transforms $$(\mbox{\boldmath$H$}f)(x)=\int^\infty_0H^{m,n}_{\thinspace p,q} \left[xt\left|\begin{array}{c}(a_i,\alpha_i)_{1,p}\\[1mm](b_j,\beta_j)_{1,q} \end{array}\right.\right]f(t)dt$$ involving Fox's $H$-functions as kernels…
Generalizing the algebra of motion-invariant differential operators on a symmetric space we study invariant operators on equivariant vector bundles. We show that the eigenequation is equivalent to the corresponding eigenequation with…
Direct and inverse source problems of a fractional diffusion equation with regularized Caputo-like counterpart hyper-Bessel operator are considered. Solutions to these problems are constructed based on appropriate eigenfunction expansion…
This paper deals with the inverse spectral problem for a non-self-adjoint Sturm-Liouville operator with discontinuous conditions inside the interval. We obtain that if the potential $q$ is known a priori on a subinterval $ \left[ b,\pi…
We consider the non-self-adjoint Sturm-Liouville operator on a finite interval. The inverse spectral problem is studied, which consists in recovering this operator from its eigenvalues and generalized weight numbers. We prove local…
We consider initial value problems for differential-algebraic equations in a possibly infinite-dimensional Hilbert space. Assuming a growth condition for the associated operator pencil, we prove existence and uniqueness of solutions for…
In this paper we investigate a discrete approximation in time and in space of a Hilbert space valued stochastic process $\{u(t)\}_{t\in [0,T]}$ satisfying a stochastic linear evolution equation with a positive-type memory term driven by an…
Consider two inverse problems for Sturm-Liouville problems on the unit interval. It means that there are two corresponding mappings $F, f$ from a Hilbert space of potentials $H$ into their spectral data. They are called isomorphic if $F$ is…
Let $L_0$ be a closed densely defined symmetric semi-bounded operator with nonzero defect indexes in a separable Hilbert space ${\cal H}$. With $L_0$ we associate a metric space $\Omega_{L_0}$ that is named a {\it wave spectrum} and…
Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on $L^{p}(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of…
In a finite-dimensional Euclidian space we consider a connected metric graph with the following property: each two cycles can have at most one common point. Such graphs are called A-graphs. On noncompact A-graph we consider a scattering…
In this paper, we consider the recovery of third-order differential operators from two spectra, as well as fourth-order or fifth-order differential operators from three spectra, where these differential operators are endowed with…
The initial value problem and global properties of solutions are studied for the vector equation: $\Big(\|u'\|^{l}u'\Big)'+\|A^{\frac{1}{2}}u\|^\beta Au+g(u')=0$ in a finite dimensional Hilbert space under suitable assumptions on $g$.
The abstract issue of 'Krylov solvability' is extensively discussed for the inverse problem $Af = g$ where $A$ is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and $g$ is a datum in the range of $A$. The…
Integral operators of Abel type of order a > 0 arise naturally in a large spectrum of physical processes. Their inversion requires care since the resulting inverse problem is ill-posed. The purpose of this work is to devise and analyse a…