Multivalued Linear Operators Research Articles

On solutions of differential inclusions in homogeneous spaces of functions

In this paper we study the solvability of some classes of differential inclusions with multivalued linear operators in homogeneous spaces of functions. These spaces include a large number of functional spaces like periodic functions and Bohr and Stepanov almost periodic functions. As an application, we consider some existence results for feedback control systems governed by degenerate differential equations of Sobolev type in a Banach space.

Characterisations of open multivalued linear operators

Characterisations of open multivalued linear operators

A Random Multivalued Uniform Boundedness Principle

We present a generalization of the Uniform Boundedness Principle valid for random multivalued linear operators, i.e., multivalued linear operators taking values in the space L0(Ω,Y) of random variables defined on a probability space \((\Omega,{\mathcal{A}},P)\) with values in the Banach space Y. Namely, for a family of such operators that are continuous with positive probability, if the family is pointwise bounded with probability at least δ>0, then the operators are uniformly bounded with a probability that in each case can be estimated in terms of δ and the index of continuity of the operator. To achieve this result, we develop the fundamental theory of multivalued linear operators on general topological vector spaces. In particular, we exhibit versions of the Closed Graph Theorem, the Open Mapping Theorem, and the Uniform Boundedness Principle for multivalued operators between F-spaces.

Multivalued linear projections

A multivalued linear projection operator P defined on linear space X is a multivalued linear operator which is idempotent and has invariant domain. We show that a multivalued projection can be characterised in terms of a pair of subspaces and then establish that the class of multivalued linear projections is closed under taking adjoints and closures. We apply the characterisations of the adjoint and completion of a projection together with the closed graph and closed range theorems to give criteria for the continuity of a projection defined on a normed linear space. A new proof of the theorem on closed sums of closed subspaces in a Banach space (cf. Mennicken and Sagraloff [9, 10]) follows as a simple corollary. We then show that the topological decomposition of a space may be expressed in terms of multivalued projections. The paper is concluded with an application to multivalued semi-Fredholm relations with generalised inverses.

Lomonosov’s invariant subspace theorem for multivalued linear operators

The famous Lomonosov's invariant subspace theorem states that if a continuous linear operator T on an infinite-dimensional normed space E commutes with a compact operator K ¬= 0, i.e., TK = KT, then T has a non-trivial closed invariant subspace. We generalize this theorem for multivalued linear operators. We also provide an application to single-valued linear operators.

THE CAUCHY PROBLEM FOR A DIFFERENTIAL INCLUSION IN BANACH SPACES AND DISTRIBUTION SPACES

We study well-posedness of degenerate Cauchy problems treated as Cauchy problems for a differential inclusion with a multivalued linear operator. Using a new approach to the definition of degenerate integrated semigroups and their generators in a Banach space, we obtain a well-posedness criterion for the problem. Moreover, we consider the Cauchy problem for a differential inclusion in the space of abstract distributions and give necessary and sufficient conditions for well-posedness in the distribution space.

A functional calculus and fractional powers for multivalued linear operators

A functional calculus and fractional powers for multivalued linear operators

Approximation results for semigroups generated by multivalued linear operators and applications

We establish convergence results for sequences of multivalued linear operators $A_n$. When each $A_n$ generates a differentiable semigroup, we state conditions for convergence of the corresponding semigroups and of the solutions of linear differential inclusions governed by $A_n$. Through a method already considered in the literature, this enables us to prove new convergence properties for a sequence of degenerate evolution equations. Applications to partial differential problems are also given.

Sums of generators of analytic semigroups and multivalued linear operators

Let A and B be generators of analytic semigroups in a Banach space. Under some conditions on the commutator of the resolvents of A and B, already considered in the literature and not implying relative boundedness, we prove that the closure of A+B (or a proper extension of it) also generates an analytic semigroup, and we characterize interpolation spaces related to it. As a tool, we use approximation and interpolation results for multivalued linear operators.

Generation theorem of semigroup for multivalued linear operators

Generation theorem of semigroup for multivalued linear operators

Least-squares solutions of multi-valued linear operator equations in Hilbert spaces

Let M be a linear manifold in H 1 ⊕ H 2, where H 1, and H 2 are Hilbert spaces. Two notions of least-squares solutions for the multi-valued linear operator equation (inclusion) y ϵ M( x) are introduced and investigated. The main results include (i) equivalent conditions for least-squares solvability, (ii) properties of a least-squares solution, (iii) characterizations of the set of all least-squares solutions in terms of algebraic operator parts and generalized inverses of linear manifolds, and (iv) best approximation properties of generalized inverses and operator parts of multi-valued linear operators. The principal tools in this investigation are an abstract adjoint theory, orthogonal operator parts, and orthogonal generalized inverses of linear manifolds in Hilbert spaces.