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Characterization of continuous homomorphisms on entire slice monogenic functions

Part of: Special classes of linear operators Holomorphic functions of several complex variables

Published online by Cambridge University Press:  17 May 2024

Stefano Pinton  and
Peter Schlosser
Stefano Pinton*
Affiliation:
(SP) Politecnico di Milano, Dipartimento di Matematica, Milano, Italy
Peter Schlosser
Affiliation:
(SP) Politecnico di Milano, Dipartimento di Matematica, Milano, Italy
*
Corresponding author: Stefano Pinton, email: stefano.pinton@polimi.it
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Abstract

This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.

Keywords

infinite order differential operatorsproximate order for slice monogenic functionscontinuous homomorphisms

MSC classification

Primary: 32A15: Entire functions 32A10: Holomorphic functions 47B38: Operators on function spaces (general)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 3 , August 2024 , pp. 892 - 920
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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