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Maximum principles for matrix-valued analytic functions. (English) Zbl 1436.30001

Summary: To what extent is the maximum modulus principle for scalar-valued analytic functions valid for matrix-valued analytic functions? In response, we discuss some maximum norm principles for such functions that do not appear to be widely known, deduce maximum and minimum principles for their singular values, and make some observations concerning resolvents and matrix exponentials.

MSC:

30A10 Inequalities in the complex plane
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
15A18 Eigenvalues, singular values, and eigenvectors

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References:

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