Unbounded Linear Operators Research Articles

Dua Formula Eksponensial (Operator Linear dari Semigrup)

Assumed A is infinitesimal generator of C0-semigroup T(t) on X. This could be defined as T(t)=etA, applies if A is a bounded linear operator. Not if A is unbounded linear operator, then it will result in one possibility that show T(t) could be represented as etA. This paper will discuss and detail the proof of the other two formulas that show T(t) could be represented as etA.

AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

AbstractLet A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

Input-to-State Stability of homogeneous infinite dimensional systems with locally Lipschitz nonlinearities

The Input-to-State Stability (ISS) of homogeneous evolution equations with unbounded linear operators and locally Lipschitz nonlinearities in Banach spaces is studied using a new homogeneous converse Lyapunov theorem. It is shown that, similarly to finite-dimensional models, uniform asymptotic stability of an unperturbed homogeneous system guarantees its ISS with respect to homogeneously involved exogenous inputs.

Toeplitz Operators whose Symbols Are Borel Measures

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator T μ with symbol μ , as an unbounded linear operator on the Hardy space. We then study various properties of T μ . Among them, there is a theorem that the domain of T μ is represented by a trichotomy. Also, it was shown that if the domain of T μ contains at least one polynomial, then T μ is densely defined. In addition, we give evidence for the conjecture that T μ with a singular measure μ reduces to a trivial linear operator.

On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space

We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type multivalued nonlinearity satisfying some monotonicity condition in the phase variables. We used the Yosida approximations of the linear part of the inclusion, the method of a priori estimates of solutions, and the topological degree method for condensing vector fields. As an example, we considered the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type.

Dissipative Shallow Water Equations: a port-Hamiltonian formulation

Abstract The dissipative Shallow Water Equations (DSWEs) are investigated as port-Hamiltonian systems. Dissipation models of different types are considered: either as nonlinear bounded operators, or as linear unbounded operators involving a classical diffusion term in 1D, or the vectorial Laplacian in 2D. In order to recast the dissipative SWE into the framework of pHs with dissipation, a physically meaningful factorization of the vectorial Laplacian is being used, which nicely separates the divergent and the rotational components of the velocity field. Finally, the structure-preserving numerical scheme provided by the Partitioned Finite Element Method (PFEM) is applied to the nonlinear bounded dissipative fluid models. For the linear unbounded cases, a change of variables is highlighted, to transform the DSWEs into a new pHs with a polynomial structure, which proves more suitable for numerics.

Krylov Solvability of Unbounded Inverse Linear Problems

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 question consists of whether the solution f can be approximated in the Hilbert norm by finite linear combinations of g,Ag,A^2g,dots , and whether solutions of this sort exist and are unique. After revisiting the known picture when A is bounded, we study the general case of a densely defined and closed A. Intrinsic operator-theoretic mechanisms are identified that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques.

Semilinear fractional-order evolution equations of Sobolev type in the sectorial case

ABSTRACT The local unique solvability of the Cauchy-type problem to a semilinear equation in a Banach space, which is solved with respect to the highest order Riemann–Liouville derivative, is proved. A linear unbounded operator at the unknown function in the equation generates an analytic in a sector resolving the family of operators of the linear homogeneous fractional-order equation. This result is applied to the study of initial-boundary value problems for a class of nonlinear partial differential equations, in particular, containing a nonlinear superdiffusion equation. Besides, it is used for the investigation of the local unique solvability of the Showalter–Sidorov type problem to a semilinear Sobolev-type equation in a Banach space with a sectorial pair of operators and with Riemann–Liouville derivatives. For this aim, we essentially use two types of condition on the nonlinear operator: the condition on the image of this operator or the condition of its independence of the elements of the degeneration subspace. These abstract results are demonstrated on examples of initial-boundary value problems for Sobolev-type nonlinear partial differential equations.

Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer

We study a dynamic boundary condition problem in heat transfer which represents the interaction between a conducting solid enclosed by a conducting shell. Both the solid and the shell are thermally inhomogeneous and anisotropic. Interaction is modelled by considering the solid as a source of thermal energy to the shell. A constitutive equation proposed by Carslaw and Jaeger establishes a relation between temperature in the shell and the boundary value of temperature in the solid. This gives rise to a dynamic boundary condition problem that has not been studied in the recent literature. The system of equations so obtained is presented as an implicit evolution equation which involves a pair of unbounded linear operators that map between two different spaces. We extend the operators to a jointly closed pair for which the implicit equation makes sense. The solution of the initial value problem is constructed by means of a holomorphic family of solution operators. The class of admissible initial states is surprisingly large.

On the Structure of Unbounded Linear Operators

Given a densely defined closed operator $$T:{\mathcal {D}}(T)\subset H\rightarrow K$$ , von Neumann defined $$W:=(I+T^*T)^{-1}$$ and showed that $$0\le W\le I$$ , $${\mathcal {K}}(W)=\{0\}$$ and $$T^*T=(I-W)W^{-1}=W^{-1}(I-W)$$ with $${\mathcal {D}}(T^*T)={\mathcal {R}}(W)$$ . (Here, $${\mathcal {D}}(\cdot )$$ , $${\mathcal {R}}(\cdot )$$ , $${\mathcal {K}}(\cdot )$$ and, later, $${\mathcal {G}}(T)$$ stand for the domain, the range, the kernel, and the graph of a linear transformation, respectively.) The functional calculus is not applicable, in general, to guess a formula like $$(I-W)^{1/2}W^{-1/2}$$ for $$|T|(:=(T^*T)^{1/2})$$ and to achieve a polar decomposition $$T=V|T|$$ for T. Also, the operators $${\bar{T}}$$ and $$T^*$$ do not exist as single-valued operators to be able to define W and extend our conjectures to arbitrary unbounded linear operators. The task of the present paper is to define the von Neumann operator W directly from $${\mathcal {G}}(T)$$ and prove all the desired extensions.

Almost periodic solutions of periodic second order linear evolution equations

The paper is concerned with periodic linear evolution equations of the form $x''(t)=A(t)x(t)+f(t)$, where $A(t)$ is a family of (unbounded) linear operators in a Banach space $X$, strongly and periodically depending on $t$, $f$ is an almost (or asymptotic) almost periodic function. We study conditions for this equation to have almost periodic solutions on ${\mathbb R}$ as well as to have asymptotic almost periodic solutions on ${\mathbb R}^+$. We convert the second order equation under consideration into a first order equation to use the spectral theory of functions as well as recent methods of study. We obtain new conditions that are stated in terms of the spectrum of the monodromy operator associated with the first order equation and the frequencies of the forcing term $f$.

M-Isometric operators and their local properties

In this paper we give necessary and sufficient conditions for a bounded linear Hilbert space operator to be an m-isometry for an unspecified m written in terms of conditions that are applied to “one vector at a time”. We provide criteria for orthogonality of generalized eigenvectors of an (a priori unbounded) linear operator T on an inner product space that correspond to distinct eigenvalues of modulus 1. We also discuss a similar question of when Jordan blocks of T corresponding to distinct eigenvalues of modulus 1 are “orthogonal”.

The essential numerical range for unbounded linear operators

We introduce the concept of essential numerical range We(T) for unbounded Hilbert space operators T and study its fundamental properties including possible equivalent characterizations and perturbation results. Many of the properties known for the bounded case do not carry over to the unbounded case, and new interesting phenomena arise which we illustrate by some striking examples. A key feature of the essential numerical range We(T) is that it captures spectral pollution in a unified and minimal way when approximating T by projection methods or domain truncation methods for PDEs.

Regular propagators of bilinear quantum systems

The present analysis deals with the regularity of solutions of bilinear control systems of the type x′=(A+u(t)B)x where the state x belongs to some complex infinite dimensional Hilbert space, the (possibly unbounded) linear operators A and B are skew-adjoint and the control u is a real valued function. Such systems arise, for instance, in quantum control with the bilinear Schrödinger equation. For the sake of the regularity analysis, we consider a more general framework where A and B are generators of contraction semigroups.Under some hypotheses on the commutator of the operators A and B, it is possible to extend the definition of solution for controls in the set of Radon measures to obtain precise a priori energy estimates on the solutions, leading to a natural extension of the celebrated noncontrollability result of Ball, Marsden, and Slemrod in 1982.

A remark on a paper of P. B. Djakov and M. S. Ramanujan

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\{o}the matrices when every continuous linear operator between l-K\{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

Sequence of Linear Operators in Non-Archimedean Banach Spaces

The aim of this paper is to find a non-Archimedean counterpart of the generalized convergence of closable unbounded linear operators as defined by Kato (Perturbation Theory for Linear Operators, 2nd edn. In: Grundlehren der Mathematischen Wissenschaften, Band 132, Springer, Berlin, 1976). Moreover, we prove that this convergence can be considered as a generalization of convergence in norm for unbounded linear operators on non-Archimedean Banach spaces (see Theorem 3.8).

A Distributed Control Problem for a Fractional Tumor Growth Model

In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three self-adjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn–Hilliard type phase field system modeling tumor growth that has been proposed by Hawkins–Daarud, van der Zee and Oden. The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in a recent work by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

Solution Theory, Variational Formulations, and Functional a Posteriori Error Estimates for General First Order Systems with Applications to Electro-Magneto-Statics and More

We prove a comprehensive solution theory using tools from functional analysis, show corresponding variational formulations, and present functional a posteriori error estimates for general linear first order systems of typefor two densely defined and closed (possibly unbounded) linear operators A1 and A2 having the complex property . As a prototypical application we will discuss the system of electro-magneto statics in 3D with mixed tangential and normal boundary conditionsOur theory covers a lot more applications in 2D, 3D, and ND, such as general differential forms and all kind of systems arising, e.g., in general relativity, biharmonic problems, Stokes equations, or linear elasticity, to mention just a few, for exampleall with possibly mixed boundary conditions of generalized tangential and normal type. Second order systems of typeswill be considered as well using the same techniques.

Determination of the parameter of the differential equation of fractional order with the Caputo derivative in Hilbert space

We consider an abstract differential equation of the first order with unbounded linear operator Dαu(t) = Au(t) + f(t) in a Hilbert space H, where Dα is the Caputo fractional derivative. For this equation the Cauchy problem is studied with initial data u(0) = u0 at t = 0. We assume that the operator A is self-adjoint and non-positive. The inverse problem to determine the nonhomogeneous member is considered under the assumption that this term has the following representation f(t) = φ(t)p, where the scalar function φ(t) is given and the element p is an unknown element of the space H. This problem belongs to the class of inverse problems. Inverse problems for abstract differential equations was discussed initially with this inhomogeneous structure member but for equations with an operator generating a C0-semigroup and usual derivative. An additional condition u(T ) = u1 is specified for the determination of the unknown p, where u1 is a given element of the space H. Thus we get the two-point problem which is only beginning to be studied for the case of fractional derivatives. The question of existence and uniqueness of the classical solution of the inverse problem is studied. Sufficient conditions of correct solvability of the inverse problem are obtained. Explicit formula to determine the unknown element in the differential equation is given.

Shift-invert Rational Krylov method for an operator $$\phi $$ ϕ -function of an unbounded linear operator

The product of a matrix function and a vector is used to solve evolution equations numerically. Hashimoto and Nodera (ANZIAM J 58:C149–C161, 2016) proposed the Shift-invert Rational Krylov method for computing these products. However, since matrices produced by evolution equations behave like unbounded operators in infinite-dimensional spaces, an analysis with the unbounded operator is essential. In this paper, the Shift-invert Rational Krylov method is extended to be applied to unbounded operators.