Positive matrix $H^{1/2}$ and Hermitian matrix $H^{1}$ functions are constant
Authors:
H. Salehi and G. D. Taylor
Journal:
Proc. Amer. Math. Soc. 26 (1970), 469-470
MSC:
Primary 30.67
DOI:
https://doi.org/10.1090/S0002-9939-1970-0267106-2
MathSciNet review:
0267106
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Abstract | References | Similar Articles | Additional Information
Abstract: It is known that if $f \in {H^{1/2}}$ and $f(z) \geqq 0$ a.e. for $|z| = 1$ then $f(z)$ is a constant. Also if $f \in {H^1}$ and $f(z)$ is real a.e. for $|z| = 1$, then $f$ is a constant. In this note we extend these two results to matrix-valued functions.
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P. L. Duren, Lecture notes on ${H^p}$ spaces, Univ. of Michigan, Ann Arbor, 1967.
- Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
- J. Neuwirth and D. J. Newman, Positive $H^{1/2}$ functions are constants, Proc. Amer. Math. Soc. 18 (1967), 958. MR 213576, DOI https://doi.org/10.1090/S0002-9939-1967-0213576-5
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Additional Information
Keywords:
Positive matrix functions,
hermitian matrix functions,
Hardy classes of functions
Article copyright:
© Copyright 1970
American Mathematical Society