# Pointwise axiomatic spectral theory in Banach algebras

#### by Raymond Lubansky

Institution: The University of Melbourne
Date: Wed 28th November 2007
Time: 2:15 PM
Location: Room 213, Richard Berry Building, The Uni of Melbourne

Abstract: Spectral theory is the study of a generalised form of eigenvalues and
invertibility in Banach algebras and in particular, the spectral mapping
theorem which in matrix theory states that for a matrix A and a
polynomial p, the eigenvalues of p(A) are exactly the set p(z) where z
is an eigenvalue of A.

Axiomatic spectral theory is the generalisation of spectral theory to
other types of pseudo-invertibility and the associated spectral mapping
theorem and has applications to various areas such as differential
equations.

The seminar will consist of a gentle introduction to Banach algebras and
spectral theory through matrix and function algebras; followed by a
display of the current state of axiomatic spectral theory and a
demonstration of pointwise spectral theory. The talk will be guided by
the spectral mapping theorem, from its polynomial form for eigenvalues
to its most general form with holomorphic functions on the spectrum.