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Transactions of the American Mathematical Society

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Invariant subspaces of the Dirichlet shift and pseudocontinuations


Authors: Stefan Richter and Carl Sundberg
Journal: Trans. Amer. Math. Soc. 341 (1994), 863-879
MSC: Primary 47B37; Secondary 30H05, 47A15
DOI: https://doi.org/10.1090/S0002-9947-1994-1145733-9
MathSciNet review: 1145733
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Abstract: In this paper we study extremal functions for invariant subspaces $ \mathcal{M}$ of the Dirichlet shift, i.e., solutions $ \varphi $ of the extremal problem

$\displaystyle \sup \{ \vert{f^{(n)}}(0)\vert/{\left\Vert f \right\Vert _D}:f \in \mathcal{M},f \ne 0\} $

. Here $ n$ is the smallest nonnegative integer such that the sup is positive. It is known that such a function $ \varphi $ generates $ \mathcal{M}$. We show that the derivative $ (z\varphi )\prime $ has a pseudocontinuation to the exterior disc. This pseudocontinuation is an analytic continuation exactly near those points of the unit circle where $ \varphi $ is bounded away from zero. We also show that the radial limit of the absolute value of an extremal function exists at every point of the unit circle. Some of our results are valid for all functions that are orthogonal to a nonzero invariant subspace.

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

  • [1] L. Brown and A. Shields, Cyclic vectors in the Dirichlet space, Trans. Amer. Math. Soc. 285 (1984), 269-304. MR 748841 (86d:30079)
  • [2] L. Carleson, Sets of uniqueness for functions regular in the unit circle, Acta. Math. 87 (1952), 325-345. MR 0050011 (14:261a)
  • [3] -, On the zeros of functions with bounded Dirichlet integrals, Math. Z. 56 (1952), 289-295. MR 0051298 (14:458e)
  • [4] -, Selected problems on exceptional sets, Van Nostrand, Princeton, N.J., 1967. MR 0225986 (37:1576)
  • [5] J. Douglas, Solution of the problem of Plateau, Trans. Amer. Math. Soc. 33 (1931), 263-321. MR 1501590
  • [6] R. Douglas, H. Shapiro, and A. Shields, Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20 (1970), 37-76. MR 0270196 (42:5088)
  • [7] P. Duren, Theory of $ {H^p}$-spaces, Pure and Appl. Math., vol. 38, Academic Press, New York, 1970. MR 0268655 (42:3552)
  • [8] J. Garnett, Bounded analytic functions, Pure and Appl. Math., vol. 96, Academic Press, New York, 1981. MR 628971 (83g:30037)
  • [9] H. Hedenmalm, A factorization theorem for square area-integrable functions, J. Reine Angew. Math. 422 (1991), 45-68. MR 1133317 (93c:30053)
  • [10] H. Hedenmalm and A. Shields, Invariant subspaces in Banach spaces of analytic functions, Michigan Math. J. 37 (1990), 91-104. MR 1042516 (91g:46059)
  • [11] J. Moeller, On the spectra of some translation invariant spaces, J. Math. Anal. Appl. 4 (1962), 276-296. MR 0150592 (27:588)
  • [12] S. Richter, Invariant subspaces in Banach spaces of analytic functions, Trans. Amer. Math. Soc. 304 (1987), 585-616. MR 911086 (88m:47056)
  • [13] -, Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988), 205-220. MR 936999 (89e:47048)
  • [14] -, A representation theorem for cyclic analytic two-isometries, Trans. Amer. Math. Soc. 328 (1991). MR 1013337 (92e:47052)
  • [15] S. Richter and A. Shields, Bounded functions in the Dirichlet space, Math. Z. 198 (1988), 151-159. MR 939532 (89c:46039)
  • [16] S. Richter and C. Sundberg, A formula for the local Dirichlet integral, Michigan Math. J. 38 (1991), 355-379. MR 1116495 (92i:47035)
  • [17] -, Multipliers and invariant subspaces of the Dirichlet shift, J. Operator Theory (to appear).
  • [18] H. Shapiro, Some remarks on weighted polynomial approximations of holomorphic functions, Math. Sb. 73(115) (1967), 320-330; English transl. in Math. USSR-Sb. 2 (1967), 285-294. MR 0217304 (36:395)
  • [19] H. Shapiro and A. Shields, On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962), 217-229. MR 0145082 (26:2617)
  • [20] J. Shapiro, Cluster set, essential range, and distance estimates in $ BMO$, Michigan Math. J. 34 (1987), 323-336. MR 911808 (89b:30029)
  • [21] D. Stegenga, A geometric condition which implies $ BMOA$, Harmonic Analysis in Euclidean Spaces. Proc. Sympos. Pure Math., vol. 35, part 1, Amer. Math. Soc., Providence, R.I., 1979, pp. 427-430. MR 545283 (80h:30033)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1145733-9
Keywords: Dirichlet space, invariant subspaces, pseudocontinuation, analytic continuation
Article copyright: © Copyright 1994 American Mathematical Society

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