## Abstract

We prove an operator-valued Laplace multiplier theorem for causal translation-invariant linear

operators which provides a characterization of continuity from H^\alpha(R,U) to H^\beta(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.

operators which provides a characterization of continuity from H^\alpha(R,U) to H^\beta(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.

Original language | English |
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Journal | Mathematics of Control, Signals, and Systems |

Early online date | 30 May 2024 |

DOIs | |

Publication status | E-pub ahead of print - 30 May 2024 |

### Data Availability Statement

No datasets were generated or analysed during the current study.## Keywords

- Causal translation-invariant operators
- Fourier transform
- Mathematical systems and control theory
- Operator-valued multipliers