A bivariate extension of the Crouzeix-Palencia result with an application to Fréchet derivatives of matrix functions

07/19/2020

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by   Michel Crouzeix, et al.

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A result by Crouzeix and Palencia states that the spectral norm of a matrix function f(A) is bounded by K = 1+√(2) times the maximum of f on W(A), the numerical range of A. The purpose of this work is to point out that this result extends to a certain notion of bivariate matrix functions; the spectral norm of f{A,B} is bounded by K^2 times the maximum of f on W(A)× W(B). As a special case, it follows that the spectral norm of the Fréchet derivative of f(A) is bounded by K^2 times the maximum of f^' on W(A). An application to the convergence analysis of certain Krylov subspace methods and the extension to functions in more than two variables are discussed.