Representations and Regularity of Vector-Valued Right-Shift Invariant Operators Between Half-Line Bessel Potential Spaces

Guiver, Chris; Opmeer, Mark R.

Abstract

Representation and boundedness properties of linear, right-shift invariant operators on half-line Bessel potential spaces (also known as fractional-order Sobolev spaces) as operator-valued multiplication operators in terms of the Laplace transform are considered. These objects are closely related to the input–output operators of linear, time-invariant control systems. Characterisations of when such operators map continuously between certain interpolation spaces and/or Bessel potential spaces are provided, including characterisations in terms of boundedness and integrability properties of the symbol, also known as the transfer function in this setting. The paper considers the Hilbert space case, and the theory is illustrated by a range of examples.

Citation

Guiver, C., & Opmeer, M. R. (2023). Representations and Regularity of Vector-Valued Right-Shift Invariant Operators Between Half-Line Bessel Potential Spaces. Integral Equations and Operator Theory, 95(3), Article 19. https://doi.org/10.1007/s00020-023-02738-3

• Journal Article Type: Article
• Acceptance Date: Jul 14, 2023
• Online Publication Date: Aug 25, 2023
• Publication Date: 2023-09
• Deposit Date: Jul 10, 2023
• Publicly Available Date: Jul 11, 2023
• Print ISSN: 0378-620X
• Electronic ISSN: 1420-8989
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 95
• Issue: 3
• Article Number: 19
• DOI: https://doi.org/10.1007/s00020-023-02738-3
• Keywords: Bessel potential space, fractional-order Sobolev space, inputoutput operator, interpolation space, Laplace transform, mathematical systems and control theory, multiplier theorem, Paley-Wiener Theorem, shift-invariant operator, Wiener-Hopf integral operator