Functional calculus for diagonalizable matrices

TL;DR

This paper characterizes when the functional calculus for diagonalizable matrices is continuous, showing it depends on the smoothness or holomorphic nature of the function f, with different conditions for complex and real domains.

Contribution

It provides a complete characterization of the continuity of the matrix functional calculus based on the regularity of the function f for various domains and matrix dimensions.

Findings

fop is continuous iff f is holomorphic for open domains in C when k > 2.

fop is continuous iff f is C^{k-2} and locally Lipschitz on intervals in R for k > 2.

Full characterization of continuity for arbitrary domains and infinite-dimensional matrices.

Abstract

For an arbitrary function f:\Omega \rightarrow C (where \Omega is a subset of the field C) and a positive integer k let f act on all diagonalizable complex matrices whose all eigenvalues lie in Omega in the following way: f[P Diag(z1,...,zk) P-1] = P Diag(f(z1),...,f(zk)) P-1 for arbitrary numbers z1,...,zk in \Omega and an invertible k \times k matrix P. The aim of the paper is to fully answer the question of when the function fop defined above is continuous for fixed k. In particular, it is shown that if \Omega is open in C, then fop is continuous for fixed k > 2 iff f is holomorphic; and if \Omega is an interval in R and k > 2, then fop is continuous iff f is of class Ck-2(\Omega) and f(k-2) is locally Lipschitz in \Omega. Also a full characterization is given when the domain of f is arbitrary as well as when fop acts on infinite-dimensional (diagonalizable) matrices.