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A nonprincipal invariant subspace of the Hardy space on the torus

Author: Chester Alan Jacewicz
Journal: Proc. Amer. Math. Soc. 31 (1972), 127-129
MSC: Primary 32.12
MathSciNet review: 0287025
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Abstract: Let $ H^2(U^n)$ be the usual Hardy space (with index 2) of holomorphic functions on $ U^n$, the unit polydisc in complex $ n$-space. A subspace of $ H^2(U^n)$ is invariant if closed under multiplication by the coordinate functions. To solve a problem left open in a paper of P. R. Ahem and D. N. Clark and a book by W. Rudin the author constructs a closed invariant subspace $ M$ of $ H^2(U^2)$ with (1) an $ f$ in $ M$ never vanishing on $ U^2$ and (2) each $ g$ in $ M$ being contained in a proper closed invariant subspace of $ M$. This easily extends to $ n \geqq 2$.

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

  • [1] P. R. Ahern and D. N. Clark, Invariant subspaces and analytic continuation in several variables., J. Math. Mech. 19 (1969/1970), 963–969. MR 0261340
  • [2] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
  • [3] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
  • [4] Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841

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Keywords: Hardy space, invariant subspace, outer function, multiple Fourier series, half-plane, semigroup of characters
Article copyright: © Copyright 1972 American Mathematical Society