Geometria, Algebra e loro applicazioni | From the bounded $S$-functional calculus to the $H^∞$-functional calculus

19 Febbraio 2025 17:00 CET Zoom online

Peter Schlosser Graz University of Technology, Austria

From the bounded $S$-functional calculus to the $H^∞$-functional calculus

Parole chiave

Abstract The quaternionic (or Clifford algebra) version of the Cauchy integral formula

$\displaystyle f(p) = \dfrac{1}{2\pi}\int_{\partial U\cap \mathbb{C}_{J}} S_L^{-1}(s,p) \mathrm{d}s_J f(s),\qquad\qquad(1)$

motivates the so called bounded $S$-functional calculus. I.e., we formally replace the variable $p$ by a bounded operator $T$ in (1), and get

$\displaystyle f(T) = \dfrac{1}{2\pi}\int_{\partial U\cap \mathbb{C}_{J}} S_L^{-1}(s,T) \mathrm{d}s_J f(s).\qquad\qquad(2)$

For bounded operators $T$, the integral (2) makes sense, since the spectrum of $T$ is bounded and the integration path $\partial U\cap \mathbb{C}_{J}$ is compact.

For unbounded, closed operators $T$, there are different generalizations of the functional calculus (2). In particular, for the important class of sectorial operators, this talk will introduce the $H^\infty$-functional calculus in a 2-step procedure. First, functions $f$ with a certain decay at $\infty$ are considered, enough decay such that the integral (2) still makes sense. In a second step, using a regularization procedure, $f(T)$ is defined also for polynomially growing functions.