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Transactions of the American Mathematical Society

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Isometries of $ \sp{\ast} $-invariant subspaces


Author: Arthur Lubin
Journal: Trans. Amer. Math. Soc. 190 (1974), 405-415
MSC: Primary 47B37; Secondary 47A15
DOI: https://doi.org/10.1090/S0002-9947-1974-0338823-9
MathSciNet review: 0338823
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Abstract: We consider families of increasing $ ^\ast$-invariant subspaces of $ {H^2}(D)$, and from these we construct canonical isometrics from certain $ {L^2}$ spaces to $ {H^2}$. We give necessary and sufficient conditions for these maps to be unitary, and discuss the relevance to a problem concerning a concrete model theory for a certain class of operators.


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

  • [1] P. R. Ahern and D. N. Clark, On functions orthogonal to invariant subspaces, Acta. Math. 124 (1970), 191-204. MR 41 #8981a. MR 0264385 (41:8981a)
  • [2] A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta. Math. 81 (1948), 17 pp. MR 10, 381. MR 0027954 (10:381e)
  • [3] N. Dunford and J. T. Schwartz, Linear operators. II. Spectral theory. Selfadjoint operators in Hilbert space, Interscience, New York, 1963. MR 32 #6181. MR 0188745 (32:6181)
  • [4] F. Hausdorff, Set theory, 2nd ed., Chelsea, New York, 1962. MR 25 #4999. MR 0141601 (25:4999)
  • [5] K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 24 #A2844. MR 0133008 (24:A2844)
  • [6] T. L. Kriete, A generalized Paley-Wiener theorem, J. Math. Anal. Appl. 36 (1971), 529-555. MR 44 #5473. MR 0288275 (44:5473)
  • [7] -, Fourier transforms and chains of inner functions, Duke Math. J. 40 (1973). MR 0328659 (48:7001)
  • [8] K. Kuratowski, Topologie. Vol. I, 2nd ed., Monografie Mat., Tom 20, Warszawa-Wrocław, 1948; English transl., new ed., rev. and aug., Academic Press, New York; PWN, Warsaw, 1966. MR 10, 389; 36 #840. MR 0217751 (36:840)
  • [9] A. Lubin, Extensions of measures and the von Neumann selection theorem, Proc. Amer. Math. Soc. 43 (1974), 118-122. MR 0330393 (48:8730)
  • [10] -, Isometries of $ ^\ast$-invariant subspaces of $ {H^2}(D)$, Thesis, University of Wisconsin, Madison, Wis., 1972.
  • [11] J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 20, Amer. Math. Soc., Providence, R. I., 1960. MR 36 #1672a. MR 0218587 (36:1672a)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0338823-9
Keywords: Invariant subspace, inner function, restricted shift operator, concrete model theory for operators
Article copyright: © Copyright 1974 American Mathematical Society

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