Brangesian spaces in the polydisk

Author:
Dinesh Singh

Journal:
Proc. Amer. Math. Soc. **110** (1990), 971-977

MSC:
Primary 46E20; Secondary 32A10, 47A15, 47B38

DOI:
https://doi.org/10.1090/S0002-9939-1990-1028289-2

MathSciNet review:
1028289

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we extend to the polydisk a theorem of L. de Branges which characterizes the class of all Hilbert spaces that are contractively contained in the classical Hardy space of the disk and which are invariant under the shift acting as an isometry. Our theorem characterizes Hilbert spaces which are vector subspaces of and which are invariant under the operators of multiplication by the coordinate functions whose actions are isometric and which doubly commute. We do not use contractivity.

**[1]**P. Beurling,*On two problems concerning linear transformations in Hilbert space*, Acta Math.**81**(1949), 239-255. MR**0027954 (10:381e)****[2]**L. de Branges,*Square summable power series*(to appear).**[3]**P. R. Halmos,*Shifts on Hilbert spaces*, J. Reine Angew. Math.**208**(1961), 102-112. MR**0152896 (27:2868)****[4]**K. Hoffman,*Banach spaces of analytic functions*, Prentice-Hall, Englewood Cliffs, NJ, 1962. MR**0133008 (24:A2844)****[5]**P. Lax,*Translation invariant spaces*, Acta Math.**101**(1969), 163-178. MR**0105620 (21:4359)****[6]**V. Mandrekar,*The validity of Beurling theorems in polydisks*, Proc. Amer. Math. Soc.**103**, 145-148. MR**938659 (90c:32008)****[7]**J. Radlow,*Closed ideals in square summable power series*, Proc. Amer. Math. Soc.**38**(1973), 293-297. MR**0312254 (47:816)****[8]**W. Rudin,*Function theory in polydiscs*, Benjamin, New York, 1969. MR**0255841 (41:501)****[9]**D. Sarason,*Shift invariant spaces from the Brangesian viewpoint*(Proc. Sympos. on the Occasion of the Proof of the Bierbach Conjecture, 1986), Math. Surveys Monogr., Amer. Math. Soc., Providence, RI, 1986, pp. 153-166. MR**875239 (88d:47014a)****[10]**-,*Function theory on the unit circle*, Virginia Polytech. Inst. and State Univ., Blacksburg, VA, 1979.**[11]**U. N. Singh and Dinesh Singh,*An extension of a theorem of de Branges*(to appear).**[12]**M. Slocinski,*On the Wold type decomposition of a pair of commuting isometries*, Ann. Polon. Math.**37**(1980), 255-262. MR**587496 (83e:47031)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1028289-2

Keywords:
Polydisk,
de Branges's theorem

Article copyright:
© Copyright 1990
American Mathematical Society