Unbounded Linear Operators Research Articles

Pseudo-almost Periodic and Pseudo-almost Automorphic Solutions to Evolution Equations in Hilbert Spaces

In this work we prove some existence and uniqueness results for pseudo-almost periodic and pseudo-almost automorphic solutions to a class of semi-linear differential equations in Hilbert spaces using theoretical measure theory. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle and the method of the invariant subspaces for unbounded linear operators. A few illustrative examples will be discussed at the end of the paper.

On condition numbers of spectral operators in a hilbert space

We consider a linear unbounded operator \(A\) in a separable Hilbert space. with the following property: there is a normal operator \(D\) with a discrete spectrum, such \(\Vert A-D\Vert <\infty \). Besides, all the Eigen values of \(D\) are different. Under certain assumptions it is shown that \(A\) is similar to a normal operator and a sharp bound for the condition number is suggested. Applications of that bound to spectrum perturbations and operator functions are also discussed. As an illustrative example we consider a non-selfadjoint differential operator.

On Elliptic Equations with General Non-Local Boundary Conditions in UMD Spaces

In this work, we give new results concerning existence, uniqueness and maximal regularity of the strict solution of a class of elliptic equations with non-local boundary conditions containing an unbounded linear operator. This study is performed in the framework of UMD Banach spaces.

Strong solutions and the initial data space for some non-uniformly parabolic equations

The solution for this problem can be considered in different meanings, for example, a classical solution, strong solution, weak (or mild) solution, entropy solution, etc. This paper is devoted to a strong solution for this problem. If A is an unbounded linear operator in Hilbert space H, then it is proved that there does not exist a strong solution for all u0 ∈ H (see [2, 11]). It is proved in [13] that there exists a strong solution if and only if u0 ∈ [D(A), H]1/2, where [D(A), H]1/2 is an interpolation space (see [9,15]). If A is a nonlinear operator, the weak solutions of these problems is often investigated. We consider the case where the operator A has the form:

Abel averages and holomorphically pseudo-contractive maps in Banach spaces

A class of maps in a complex Banach space is studied, which includes both unbounded linear operators and nonlinear holomorphic maps. The defining property, which we call pseudo-contractivity, is introduced by means of the Abel averages of such maps. We show that the studied maps are dissipative in the spirit of the classical Lumer–Phillips theorem. For pseudo-contractive holomorphic maps, we establish the power convergence of the Abel averages to holomorphic retractions.

A non-local boundary value problem method for semi-linear parabolic equations backward in time

The ill-posed semi-linear parabolic equation backward in time with the positive self-adjoint unbounded linear operator A and being given is regularized by the well-posed non-local boundary value problem Under the condition , a priori and a posteriori parameter choice rules are suggested which yield the error estimate for some positive constant C.

A BOUND FOR SIMILARITY CONDITION NUMBERS OF UNBOUNDED OPERATORS WITH HILBERT–SCHMIDT HERMITIAN COMPONENTS

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of $H$ is a compact operator and $H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that $H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.

Well-posedness of delay parabolic equations with unbounded operators acting on delay terms

In the present paper, the well-posedness of the initial value problem for the delay differential equation d v ( t ) d t +Av(t)=B(t)v(t−ω)+f(t), t≥0; v(t)=g(t) (−ω≤t≤0) in an arbitrary Banach space E with the unbounded linear operators A and B(t) in E with dense domains D(A)⊆D(B(t)) is studied. Two main theorems on well-posedness of this problem in fractional spaces E α are established. In practice, the coercive stability estimates in Hölder norms for the solutions of the mixed problems for delay parabolic equations are obtained.MSC:35G15.

Commutativity up to a factor for bounded and unbounded operators

In this paper, we further investigate the problem of commutativity up to a factor (or λ-commutativity) in the setting of bounded and unbounded linear operators in a complex Hilbert space. The results are based on a new approach to the problem. We finish the paper by a conjecture on the commutativity of self-adjoint operators.

Bernstein-Jackson-type inequalities and Besov spaces associated with unbounded operators

Besov-type interpolation spaces and appropriate Bernstein-Jackson inequalities, generated by unbounded linear operators in a Banach space, are considered. In the case of the operator of differentiation these spaces and inequalities exactly coincide with the classical ones. Inequalities are applied to a best approximation problem in a Banach space, particularly, to spectral approximations of regular elliptic operators. MSC:47A58, 41A17.

Relative essential spectra involving relative demicompact unbounded linear operators

In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we establish some new results in Fredholm theory. Moreover, we apply the obtained results to discuss the incidence of some perturbation results on the behavior of relative essential spectra of unbounded linear operators acting on Banach spaces. We conclude by characterizations of the relative Schechter's and approximate essential spectrum.

Local attractivity in nonautonomous semilinear evolution equations

AbstractWe study the local attractivity of mild solutions of equations in the form u’(t) = A(t)u(t) + f (t, u(t)), where A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at infinity

Generalizations of Kaplansky's Theorem Involving Unbounded Linear Operators

Generalizations of Kaplansky's Theorem Involving Unbounded Linear Operators

On different concepts of closedness of linear operators

The purpose of this paper is to introduce, by means of the extensions of almost closed operators, the notion of almost closable linear operator acting in a Hilbert or Banach space. This class of operators is strictly included in the class of all unbounded linear operators, it contains the set of all closable operators and that of all almost closed operators and is invariant under finite and countable sums, finite products, limits and integrals. We also present some fundamental properties relative to almost closability and we define a locally convex Hausdorff topology in the set of all almost closable operators. Mathematics subject classification (2010): 47A05, 47B33.

Positive Steady States of Evolution Equations with Finite Dimensional Nonlinearities

We study the question of existence of positive steady states of nonlinear evolution equations. We recast the steady state equation in the form of eigenvalue problems for a parametrised family of unbounded linear operators, which are generators of strongly continuous semigroups; and a fixed point problem. In case of irreducible governing semigroups we consider evolution equations with non-monotone nonlinearities of dimension two, and we establish a new fixed point theorem for set-valued maps. In case of reducible governing semigroups we establish results for monotone nonlinearities of any finite dimension $n$. In addition, we establish a non-quasinilpotency result for a class of strictly positive operators, which are neither irreducible nor compact, in general. We illustrate our theoretical results with examples of partial differential equations arising in structured population dynamics. In particular, we establish existence of positive steady states of a size-structured juvenile-adult and a structured consumer-resource population model, as well as for a selection-mutation model with distributed recruitment process.

Deformation of striped patterns by inhomogeneities

AbstractWe study the effects of adding a local perturbation in a pattern‐forming system, taking as an example the Ginzburg–Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a periodic pattern, one finds an unbounded linear operator that is not Fredholm due to continuous spectrum in typical translation invariant or weighted spaces. We show that Kondratiev spaces, which encode algebraic localization that increases with each derivative, provide an effective means to circumvent this difficulty. We establish Fredholm properties in such spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. We find a logarithmic phase correction, which vanishes for a particular spatial shift only, which we interpret as a phase‐selection mechanism through the inhomogeneity. Copyright © 2013 John Wiley &amp; Sons, Ltd.

On regularizers of unbounded linear operators in Banach spaces

Regularizers of densely defined unbounded linear operators in Banach spaces and their applications to spectral theory are considered. Necessary and sufficient conditions in terms of regularizer properties for an unbounded operator T to have discrete spectrum are obtained. In the case where T has a self-adjoint regularizer in some Schatten--von Neumann ideals, asymptotic properties of the eigenvalues are investigated; in particular, it is shown that the eigenvalues of T asymptotically belong to some angle in the complex plane. Bibliography: 16 titles.

On the M-essential spectra of two-group transport equations

In this paper, we deal with the M-essential spectra of unbounded linear operators in Banach spaces where some generalizations of earlier work are given. Furthermore, we give an application from transport theory. Copyright © 2013 John Wiley & Sons, Ltd.

On convergence of difference schemes for delay parabolic equations

The convergence of difference schemes for the approximate solutions of the initial–boundary value problem for the delay parabolic differential equation {dv(t)dt+Av(t)=B(t)v(t−ω),t≥0,v(t)=g(t)(−ω≤t≤0) in an arbitrary Banach space E with the unbounded linear operators A and B(t) in E with dense domains D(A)⊆D(B(t)) is investigated. Theorems on convergence estimates for the solutions of the first and the second order of accuracy difference schemes in fractional spaces Eα are established. In practice, the convergence estimates in Hölder norms for the solutions of difference schemes of the first and the second order of approximation in t of the approximate solutions of multi-dimensional delay parabolic equations are obtained. The theoretical statements for the solution of these difference schemes are supported by the numerical example.

A linear regularization scheme for inverse problems with unbounded linear operators on Banach spaces

This paper extends the linear regularization scheme known as the approximate inverse to unbounded linear operators on Banach spaces. The principle of feature reconstruction is adapted from bounded operators to the unbounded scenario and, in addition, a new situation is examined where the data need to be pre-processed to fit into the mathematical model. In all these cases, invariance and regularization properties are surveyed and established for the example of fractional differentiation. Numerical results confirm the derived characteristics of the presented methods.