Quadratic estimates for the $H^\infty$-functional calculus of bisectorial Clifford operators

TL;DR

This paper establishes quadratic estimates that characterize the boundedness of the $H^ $-functional calculus for bisectorial Clifford operators in Hilbert spaces, extending classical spectral theory into hypercomplex settings.

Contribution

It provides the first characterization of the $H^ $-functional calculus boundedness for bisectorial Clifford operators via quadratic estimates, addressing challenges from hypercomplex spectral definitions.

Findings

Quadratic estimates are established for bisectorial Clifford operators.

The paper extends the $H^ $-functional calculus theory to hypercomplex operators.

It highlights differences in spectral definitions affecting proof techniques.

Abstract

The $H^\infty$-functional calculus is a two-step procedure, introduced by A. McIntosh, that allows the definition of functions of sectorial operators in Banach spaces. It plays a crucial role in the spectral theory of differential operators, as well as in their applications to evolution equations and various other fields of science. An extension of the $H^\infty$-functional calculus also exists in the hypercomplex setting, where it is based on the notion of $S$-spectrum. Originally this was done for sectorial quaternionic operators, but then also generalized all the way to bisectorial fully Clifford operators. In the latter setting and in Hilbert spaces, this paper now characterizes the boundedness of the $H^\infty$-functional calculus through certain quadratic estimates. Due to substantial differences in the definitions of the $S$-spectrum and the $S$-resolvent operators, the proofs of quadratic estimates in this setting face additional challenges compared to the classical theory of complex operators.