# Spectral Kabbalah and the converse functional calculus

** Speaker:** Jody Trout, Dartmouth College

** Date:** April 23, 2024

** Abstract:** Self-adjointness is an important property in functional analysis, differential equations, operator algebras, noncommutative geometry, index theory, spectral geometry and quantum theory. Associated to a self-adjoint (un)bounded operator on a Hilbert space are several structures: spectral measures, Cayley transforms, one-parameter unitary groups, and the C

*-functional calculus. We show how to organize these into a 2D (and even 3D) “Kabbalahistic” type diagram. In order to complete it, we need to be able to show that there is a converse to the C*-functional calculus.