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Positive matrix $ H\sp{1/2}$ and Hermitian matrix $ H\sp{1}$ functions are constant

Authors: H. Salehi and G. D. Taylor
Journal: Proc. Amer. Math. Soc. 26 (1970), 469-470
MSC: Primary 30.67
MathSciNet review: 0267106
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Abstract: It is known that if $ f \in {H^{1/2}}$ and $ f(z) \geqq 0$ a.e. for $ \vert z\vert = 1$ then $ f(z)$ is a constant. Also if $ f \in {H^1}$ and $ f(z)$ is real a.e. for $ \vert z\vert = 1$, then $ f$ is a constant. In this note we extend these two results to matrix-valued functions.

References [Enhancements On Off] (What's this?)

  • [1] P. L. Duren, Lecture notes on $ {H^p}$ spaces, Univ. of Michigan, Ann Arbor, 1967.
  • [2] H. Helson, Lectures on invariant subspaces, Academic Press, New York, 1964. MR 30 #1409. MR 0171178 (30:1409)
  • [3] J. Neuwirth and D. J. Newman, Positive $ {H^{1/2}}$ functions are constants, Proc. Amer. Math. Soc. 18 (1967), 958. MR 35 #4436. MR 0213576 (35:4436)

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Keywords: Positive matrix functions, hermitian matrix functions, Hardy classes of functions
Article copyright: © Copyright 1970 American Mathematical Society

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