Compact Linear Operators Research Articles

Asymptotic profiles of basic reproduction number for a degenerate reaction–diffusion host–pathogen model

In this paper, the basic reproduction number (BRN) is explored in terms of a degenerate reaction–diffusion host–pathogen model. The expressions of BRN and local basic reproduction number (LBRN) for the model are calculated. Some basic properties of with respect to the general diffusion rates and the asymptotic profiles of with respect to large diffusion rates and small diffusion rates are established by introducing two compact operators and . Our results reveal that is the spectral radius of a sum of the product of additive operators and strongly positive compact linear operators with spectral radii one.

A numerical range approach to Birkhoff–James orthogonality with applications

The main aim of this paper is to provide characterizations of Birkhoff–James orthogonality (BJ-orthogonality in short) in a number of families of Banach spaces in terms of the elements of significant subsets of the unit ball of their dual spaces, which makes the characterizations more applicable. The tool to do so is a fine study of the abstract numerical range and its relation with the BJ-orthogonality. Among other results, we provide a characterization of BJ-orthogonality for spaces of vector-valued bounded functions in terms of the domain set and the dual of the target space, which is applied to get results for spaces of vector-valued continuous functions, uniform algebras, Lipschitz maps, injective tensor products, bounded linear operators with respect to the operator norm and to the numerical radius, multilinear maps, and polynomials. Next, we study possible extensions of the well-known Bhatia–Šemrl theorem on BJ-orthogonality of matrices, showing results in spaces of vector-valued continuous functions, compact linear operators on reflexive spaces, and finite Blaschke products. Finally, we find applications of our results to the study of spear vectors and spear operators. We show that no smooth point of a Banach space can be BJ-orthogonal to a spear vector of Z. As a consequence, if X is a Banach space containing strongly exposed points and Y is a smooth Banach space with dimension at least two, then there are no spear operators from X to Y. Particularizing this result to the identity operator, we show that a smooth Banach space containing strongly exposed points has numerical index strictly smaller than one. These latter results partially solve some open problems.

The Properties of the Spectrals of Fuzzy Compact Linear Operators

The objective of this paper is to present a novel approach for the a-fuzzy standardized area and its fundamental characteristics with basic attributes of fuzzy compact and fuzzy bounded linear functions. Also, it presents some of the basic attributes of the spectral of fuzzy compact linear functions in terms of theorems that clearly draw the elementary properties of these functions and the mathematical relationships between them.

Determination of unknown shear force in transverse dynamic force microscopy from measured final data

Abstract In this paper, we present a new methodology, based on the inverse problem approach, for the determination of an unknown shear force acting on the inaccessible tip of the microcantilever, which is a key component of transverse dynamic force microscopy (TDFM). The mathematical modelling of this phenomenon leads to the inverse problem of determining the shear force g ⁢ ( t ) {g(t)} acting on the inaccessible boundary x = ℓ {x=\ell} in a system governed by the variable coefficient Euler–Bernoulli equation ρ A ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ⁢ x = 0 , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ) , \rho_{A}(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx}+\kappa(x)u_{xxt})_{xx}=0,\quad(x,t)% \in(0,\ell)\times(0,T), subject to the homogeneous initial conditions and the boundary conditions u ⁢ ( 0 , t ) = u 0 ⁢ ( t ) , u x ⁢ ( 0 , t ) = 0 , ( u x ⁢ x ⁢ ( x , t ) + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x = ℓ = 0 , ( - ( r ⁢ ( x ) ⁢ u x ⁢ x + κ ⁢ ( x ) ⁢ u x ⁢ x ⁢ t ) x ) x = ℓ = g ⁢ ( t ) , u(0,t)=u_{0}(t),\quad u_{x}(0,t)=0,\quad(u_{xx}(x,t)+\kappa(x)u_{xxt})_{x=\ell% }=0,\quad\bigl{(}-(r(x)u_{xx}+\kappa(x)u_{xxt})_{x}\bigr{)}_{x=\ell}=g(t), from the final time measured output (displacement) u T ⁢ ( x ) := u ⁢ ( x , T ) {u_{T}(x):=u(x,T)} . We introduce the input-output map ( Φ ⁢ g ) ⁢ ( x ) := u ⁢ ( x , T ; g ) {(\Phi g)(x):=u(x,T;g)} , g ∈ 𝒢 {g\in\mathcal{G}} , and prove that it is a compact and Lipschitz continuous linear operator. Then we introduce the Tikhonov functional J ⁢ ( F ) = 1 2 ⁢ ∥ Φ ⁢ g - u T ∥ L 2 ⁢ ( 0 , ℓ ) 2 J(F)=\frac{1}{2}\lVert\Phi g-u_{T}\rVert_{L^{2}(0,\ell)}^{2} and prove the existence of a quasi-solution of the inverse problem. We derive a gradient formula for the Fréchet gradient of the Tikhonov functional through the corresponding adjoint problem solution and prove that it is a Lipschitz continuous functional. The results of the numerical experiments clearly illustrate the effectiveness and feasibility of the proposed approach.

Dynamical analysis for a diffusive SVEIR epidemic model with nonlinear incidences

In this article, we are concerned with a diffusive SVEIR epidemic model with nonlinear incidences. We first obtain the well-posedness of solutions for the model. Then, the basic reproduction number R_{0} and the local basic reproduction number {overline{R}}_{0}(x) are calculated, which are defined as the spectral radii of the next-generation operators. The relationship between R_{0} and {overline{R}}_{0}(x) as well as the asymptotic properties of R_{0} when the diffusive rates tend to infinity or zero is investigated by introducing two compact linear operators L_{1} and L_{2}. Using the theory of monotone dynamical systems and the persistence theory of dynamical systems, we show that the disease-free equilibrium is globally asymptotically stable when R_{0}<1, while the disease is uniformly persistent when R_{0}>1. Furthermore, in the spatially homogeneous case, by using the Lyapunov functions method and LaSalle’s invariance principle, we completely obtain that the disease-free equilibrium is globally asymptotically stable if R_{0}le 1, and the endemic equilibrium is globally asymptotically stable if R_{0}>1 and an additional condition is satisfied.

A study of fuzzy anti-λ-ideal convergent triple sequence spaces

Abstract A class with ambiguous status of its elements by a membership function that assigns to each element a grade in the close interval [ 0 , 1 ] {[0,1]} ; Lofti Zadeh introduced this idea into theory as fuzzy sets in the year of 1965. A study of fuzzy anti-normed linear spaces by Kočinac on some topological properties motivated us to work on fuzzy anti-normed triple sequence spaces with respect to ideal by using compact linear operator. Further, we prove some theorems, particularly on convergence and completeness.

A Study on Compact Operators in Locally K -Convex Spaces

In this paper we give an equivalent definition of continuous and compact linear operators by using orthogonal bases in non-archimedean locally K - convex spaces. We also show that if E is a space and F is a semi-Montel space, then every continuous linear operator T:E→F is compact.

A Class of Semilinear Parabolic Problems and Analytic Semigroups

(1) Background: This paper is devoted to the study of a class of semilinear initial boundary value problems of parabolic type. (2) Methods: We make use of fractional powers of analytic semigroups and the interpolation theory of compact linear operators due to Lions–Peetre. (3) Results: We give a functional analytic proof of the C2 compactness of a bounded regular solution orbit for semilinear parabolic problems with Dirichlet, Neumann and Robin boundary conditions. (4) Conclusions: As an application, we study the dynamics of a population inhabiting a strongly heterogeneous environment that is modeled by a class of diffusive logistic equations with Dirichlet and Neumann boundary conditions.

Spectral Properties of Compact Operators

The spectral properties of a compact operator \(T : X \longrightarrow Y\) on a normed linear space resemble those of square matrices. For a compact operator, the spectral properties can be treated fairly completely in the sense that Fredholm's famous theory of integral equations may be extended to linear functional equations \(T x -\lambda\) \(= y\) with a complex parameter \(\lambda\) . This paper has studied and investigated the spectral properties of compact operators in Hilbert spaces. The spectral properties of compact linear operators are relatively simple generalization of the eigenvalues of finite matrices. As a result, the paper has given a number of corresponding propositions and interesting facts which are used to prove basic properties of compact operators. The Fredholm theory has been introduced to investigate the solvability of linear integral equations involving compact operators.

Properties of Chmielinski-orthogonality using Kadets-Klee property

The aim of this paper is to study new results of an approximate orthogonality of Birkhoff-James techniques in real Banach space , namely Chiemelinski orthogonality (even there is no ambiguity between the concepts symbolized by orthogonality) and provide some new geometric characterizations which is considered as the basis of our main definitions. Also, we explore relation between two different types of orthogonalities. First of them orthogonality in a real Banach space and the other orthogonality in the space of bounded linear operator . We obtain a complete characterizations of these two orthogonalities in some types of Banach spaces such as strictly convex space, smooth space and reflexive space. The study is designed to give different results about the concept symmetry of Chmielinski-orthogonality for a compact linear operator on a reflexive, strictly convex Banach space having Kadets-Klee property by exploring a new type of a generalized some results with Birkhoff James orthogonality in the space of bounded linear operators. We also exhibit a smooth compact linear operator with a spectral value that is defined on a reflexive, strictly convex Banach space having Kadets-Klee property either having zero nullity or not -right-symmetric.

Periodic Solution for some Class of Linear Partial Differential Equation with infinite Delay using Semi-Fredholm perturbations

Abstract In this work, we study the existence of periodic solutions for a class of linear partial functional differential equations with infinite delay. Inspiring by an existing study, by applying the perturbation theory of semi-Fredholm operators, we introduce a suitable a priori estimate on the norm of the operator L to establish the periodicity of solutions in the case where the linear part is nondensely defined and satisfies the Hille-Yosida condition and without considering the exponential stability condition on the semigroup generated by the part of this operator on the closure of it’s domain. Moreover, in the special case where the linear part generates a strongly continuous semigroup and perturbed by a compact linear operator, we give some sufficient conditions to derive periodic solution from bounded ones. Finally, our theoretical results are illustrated by applications in both densely and nondensely cases.

The degree of ill-posedness of composite linear ill-posed problems with focus on the impact of the non-compact Hausdorff moment operator

We consider compact composite linear operators in Hilbert space, where the composition is given by some compact operator followed by some non-compact one possessing a non-closed range. Focus is on the impact of the non-compact factor on the overall behavior of the decay rates of the singular values of the composition. Specifically, the composition of the compact integration operator with the non-compact Hausdorff moment operator is considered. We show that the singular values of the composite operator decay faster than those of the integration operator, providing a first example of this kind. However, there is a gap between available lower bounds for the decay rate and the obtained result. Therefore we conclude with a discussion.

Fuzzy Convergence Sequence and Fuzzy Compact Operators on Standard Fuzzy Normed Spaces

The main purpose of this work is to introduce some types of fuzzy convergence sequences of operators defined on a standard fuzzy normed space (SFN-spaces) and investigate some properties and relationships between these concepts. Firstly, the definition of weak fuzzy convergence sequence in terms of fuzzy bounded linear functional is given. Then the notions of weakly and strongly fuzzy convergence sequences of operators are introduced and essential theorems related to these concepts are proved. In particular, if ( ) is a strongly fuzzy convergent sequence with a limit where linear operator from complete standard fuzzy normed space into a standard fuzzy normed space then belongs to the set of all fuzzy bounded linear operators . Furthermore, the concept of a fuzzy compact linear operator in a standard fuzzy normed space is introduced. Also, several fundamental theorems of fuzzy compact linear operators are studied in the same space. More accurately, every fuzzy compact linear operator is proved to be fuzzy bounded where and are two standard fuzzy normed spaces

English

The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\cal H}{\cal K}$$ (X). In this paper, on each Frechet space $${\cal H}{\cal K}$$ (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\cal H}{\cal K}$$ (X) space. It is known that every control-convergent sequence in the $${\cal H}{\cal K}$$ space always belongs to a $${\cal H}{\cal K}$$ (X) space for some X. We illustrate how to apply results for Frechet spaces $${\cal H}{\cal K}$$ (X) to control-convergent sequences in the $${\cal H}{\cal K}$$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.

Hypercyclic Translation Operators on the Algebra of Compact Operators

In this paper we characterize hypercyclic translation operators on the space of all compact linear operators on a Hilbert space H. Also, we give some sufficient condition for a related cosine operator function to be chaotic or topologically transitive

Ball intersection properties in metric spaces

The goal of the present work is to introduce new metric space techniques to study the intersection properties of families of balls. These new techniques add, on the one hand, to results due to Lindenstrauss on the extension of uniformly continuous functions and compact linear operators, and answer, on the other hand, questions raised by Aronszajn and Panitchpakdi on hyperconvex metric spaces. The present work is divided into two parts. In the first part, the proofs of our two main results on ball intersection properties in metric spaces are presented. The first main result states that for any integer n greater or equal to three, a complete and almost n-hyperconvex metric space is automatically n-hyperconvex. The second main result shows that in complete metric spaces, the property of being 4-hyperconvex is equivalent to the property of being finitely hyperconvex and that the analogues are true for externally 4-hyperconvex and weakly externally 4-hyperconvex subsets. This last result and the results proved later in this work unify the analysis of those three properties: hyperconvexity, external hyperconvexity and weak external hyperconvexity. In the second part of this work, we make the link with notions of convexity and bicombings and present applications. We extend local-to-global results on hyperconvex spaces to externally hyperconvex spaces as well as to weakly externally hyperconvex spaces. We conclude with applications of our results to the characterization of externally hyperconvex and weakly externally hyperconvex subsets of hyperconvex spaces.

Compact disjointness preserving polynomials on quasi-Banach lattices

The paper is devoted to a description of compact disjointness preserving homogeneous polynomials on quasi-Banach lattices. De Pagter and Wickstead found independently that a compact linear operator between Banach lattices is a lattice homomorphism if and only if it is representable as the sum of a norm convergent series of one-dimensional lattice homomorphisms with pairwise disjoint ranges. It is shown that this result also holds for a wider class of operators and spaces, namely, for compact disjointness preserving homogeneous polynomials acting from one quasi-Banach lattice into another. As an auxiliary result it is proved that a lattice homomorphism between quasi-Banach lattices dominated by a positive compact operator is compact.

Characterization of compact linear integral operators in the space of functions of bounded variation

Although various operators in the space of functions of bounded variation have been studied by quite a few authors, no simple necessary and sufficient conditions guaranteeing compactness of linear integral operators acting in such spaces have been known. The aim of the paper is to fully characterize the class of kernels which generate compact linear integral operators in the BV-space. Using this characterization we show that certain weakly singular and convolution operators (such as the Abel and Volterra operators), when considered as transformations of \(BV[a,b]\), are compact. We also provide a detailed comparison of those new necessary and sufficient conditions with various other conditions connected with compactness of linear (integral) operators in the space of functions of bounded variation which already exist in the literature.

The Singular Value Expansion for Arbitrary Bounded Linear Operators

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed.

Two-Lipschitz operator ideals

We introduce and investigate the concept of two-Lipschitz operator ideal between pointed metric spaces and Banach spaces. We show the basics of this new theory and we give a procedure to create a two-Lipschitz operator ideal from a linear operator ideal. We apply our result to the ideals of strongly p-summing and compact linear operator to obtain their corresponding two-Lipschitz operator ideal. Also, we establish a natural relation between two-Lipschitz and bilinear maps and show that the two-Lipschitz factorable p-dominated operators are those which are associated to the well-known p-semi-integral bilinear operators.