# 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.

*For More Information:* Raymond Lubansky R.Lubansky@ms.unimelb.edu.au