Operator-valued multiplier theorems for causal translation-invariant operators with applications to control theoretic input-output stability

Abstract

We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear operators which provides a characterization of continuity from \(H^\alpha ({\mathbb {R}},U)\) to \(H^\beta ({\mathbb {R}},U)\) (fractional U-valued Sobolev spaces, U a complex Hilbert space) in terms of a certain boundedness property of the transfer function (or symbol), an operator-valued holomorphic function on the right-half of the complex plane. We identify sufficient conditions under which this boundedness property is equivalent to a similar property of the boundary function of the transfer function. Under the assumption that U is separable, the Laplace multiplier theorem is used to derive a Fourier multiplier theorem. We provide an application to mathematical control theory, by developing a novel input-output stability framework for a large class of causal translation-invariant linear operators which refines existing input-output stability theories. Furthermore, we show how our work is linked to the theory of well-posed linear systems and to results on polynomial stability of operator semigroups. Several examples are discussed in some detail.