The holomorphic functional calculus as an operational calculus

Holomorphic Functional Calculus and Some Applications

In this note, we present some applications of complex analysis to functional analysis and operator theory. We will focus on some basic facts of holomorphic functional calculus and its applications to analyze the spectrum of operators.

Multicentric Holomorphic Calculus

An operational calculus for the Mould operad

The operad of moulds is realized in terms of an operational calculus of formal integrals (continuous formal power series). This leads to many simplifications and to the discovery of various suboperads. In particular, we prove a conjecture of the first author about the inverse image of non-crossing trees in the dendriform operad. Finally, we explain a connection with the formalism of noncommutat...

Feynman’s Operational Calculus and the Stochastic Functional Calculus in Hilbert Space

Abstract. Let A1, A2 be bounded linear operators acting on a Banach space E. A pair (μ1, μ2) of continuous probability measures on [0, 1] determines a functional calculus f 7−→ fμ1,μ2(A1, A2) for analytic functions f by weighting all possible orderings of operator products of A1 and A2 via the probability measures μ1 and μ2. For example, f 7−→ fμ,μ(A1, A2) is the Weyl functional calculus with e...

Spectrum as the Support of Functional Calculus

We investigate the new definition of analytic functional calculus in the terms of representation theory of SL 2 (R). We avoid any usage of its algebraic homomorphism property and replace it by the demand to be an intertwining operator. The related notion of spectrum and spectral mapping theorem are given. The construction is illustrated by a simple example of calculus and spectrum of non-normal...