## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Workshop/Miniconference on Functional Analysis and Optimization. Simon Fitzpatrick and John Giles, eds. Proceedings of the Centre for Mathematical Analysis, v. 20. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1988), 153 - 161

### A non-commutative joint spectral theory

#### Abstract

For certain m-tuples $a = {a_1, ... ,a_n)$ of elements $a_j$ in a unital Ba11ach algebra, we construct a joint spectrum $\gamma(a)$ and a functional calculus with a spectral mapping theorem. It is not assumed that the $a_j$ j commute but rather that they commute modulo the Jacobson radical of the algebra they generate. For matrices, this last condition is equivalent to their being simultaneously triangularizable. This work extends that of M.E. Taylor, R.F.V. Anderson, and A. Mcintosh and A. Pryde.

#### Article information

**Dates**

First available in Project Euclid:
18 November 2014

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416340111

**Mathematical Reviews number (MathSciNet)**

MR1009602

**Zentralblatt MATH identifier**

0705.47013

#### Citation

Pryde, A. J. A non-commutative joint spectral theory. Workshop/Miniconference on Functional Analysis and Optimization, 153--161, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1988. https://projecteuclid.org/euclid.pcma/1416340111