Analytic contractions, nontangential limits, and the index of invariant subspaces
HTML articles powered by AMS MathViewer
- by Alexandru Aleman, Stefan Richter and Carl Sundberg PDF
- Trans. Amer. Math. Soc. 359 (2007), 3369-3407 Request permission
Abstract:
Let $\mathcal {H}$ be a Hilbert space of analytic functions on the open unit disc $\mathbb {D}$ such that the operator $M_{\zeta }$ of multiplication with the identity function $\zeta$ defines a contraction operator. In terms of the reproducing kernel for $\mathcal {H}$ we will characterize the largest set $\Delta (\mathcal {H}) \subseteq \partial \mathbb {D}$ such that for each $f, g \in \mathcal {H}$, $g \ne 0$ the meromorphic function $f/g$ has nontangential limits a.e. on $\Delta (\mathcal {H})$. We will see that the question of whether or not $\Delta (\mathcal {H})$ has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of $M_{\zeta }$. We further associate with $\mathcal {H}$ a second set $\Sigma (\mathcal {H}) \subseteq \partial \mathbb {D}$, which is defined in terms of the norm on $\mathcal {H}$. For example, $\Sigma (\mathcal {H})$ has the property that $||\zeta ^{n}f|| \to 0$ for all $f \in \mathcal {H}$ if and only if $\Sigma (\mathcal {H})$ has linear Lebesgue measure 0. It turns out that $\Delta (\mathcal {H}) \subseteq \Sigma (\mathcal {H})$ a.e., by which we mean that $\Delta (\mathcal {H}) \setminus \Sigma (\mathcal {H})$ has linear Lebesgue measure 0. We will study conditions that imply that $\Delta (\mathcal {H}) = \Sigma (\mathcal {H})$ a.e.. As one corollary to our results we will show that if dim $\mathcal {H}/\zeta \mathcal {H} =1$ and if there is a $c>0$ such that for all $f \in \mathcal {H}$ and all $\lambda \in \mathbb {D}$ we have $||\frac {\zeta -\lambda }{1-\overline {\lambda }\zeta }f||\ge c||f||$, then $\Delta (\mathcal {H}) =\Sigma (\mathcal {H})$ a.e. and the following four conditions are equivalent: (1) $||\zeta ^{n} f||\nrightarrow 0$ for some $f \in \mathcal {H}$, (2) $||\zeta ^{n} f||\nrightarrow 0$ for all $f \in \mathcal {H}$, $f \ne 0$, (3) $\Delta (\mathcal {H})$ has nonzero Lebesgue measure, (4) every nonzero invariant subspace $\mathcal {M}$ of $M_{\zeta }$ has index 1, i.e., satisfies dim $\mathcal {M}/\zeta \mathcal {M} =1$.References
- Alexandru Aleman and Stefan Richter, Some sufficient conditions for the division property of invariant subspaces in weighted Bergman spaces, J. Funct. Anal. 144 (1997), no. 2, 542–556. MR 1432597, DOI 10.1006/jfan.1996.2998
- Alexandru Aleman, Stefan Richter, and William T. Ross, Pseudocontinuations and the backward shift, Indiana Univ. Math. J. 47 (1998), no. 1, 223–276. MR 1631561, DOI 10.1512/iumj.1998.47.1583
- Alexandru Aleman, Stefan Richter, and Carl Sundberg, The majorization function and the index of invariant subspaces in the Bergman spaces, J. Anal. Math. 86 (2002), 139–182. MR 1894480, DOI 10.1007/BF02786647
- Aleman, Alexandru, Richter, Stefan, and Sundberg, Carl, Nontangential limits in $\mathcal P^t(\mu )$-spaces and the index of invariant subspaces, preprint.
- Alexandru Aleman, Stefan Richter, and Carl Sundberg, Invariant subspaces for the backward shift on Hilbert spaces of analytic functions with regular norm, Bergman spaces and related topics in complex analysis, Contemp. Math., vol. 404, Amer. Math. Soc., Providence, RI, 2006, pp. 1–25. MR 2244001, DOI 10.1090/conm/404/07631
- C. Apostol, H. Bercovici, C. Foias, and C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. I, J. Funct. Anal. 63 (1985), no. 3, 369–404. MR 808268, DOI 10.1016/0022-1236(85)90093-X
- N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404. MR 51437, DOI 10.1090/S0002-9947-1950-0051437-7
- Hari Bercovici, Factorization theorems and the structure of operators on Hilbert space, Ann. of Math. (2) 128 (1988), no. 2, 399–413. MR 960951, DOI 10.2307/1971446
- Leon Brown, Allen Shields, and Karl Zeller, On absolutely convergent exponential sums, Trans. Amer. Math. Soc. 96 (1960), 162–183. MR 142763, DOI 10.1090/S0002-9947-1960-0142763-8
- Bernard Chevreau, George R. Exner, and Carl M. Pearcy, Boundary sets for a contraction, J. Operator Theory 34 (1995), no. 2, 347–380. MR 1373328
- John B. Conway, A course in functional analysis, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR 1070713
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- Peter Duren and Alexander Schuster, Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR 2033762, DOI 10.1090/surv/100
- John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
- Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
- Thomas Kriete and Tavan Trent, Growth near the boundary in $H^{2}(\mu )$ spaces, Proc. Amer. Math. Soc. 62 (1976), no. 1, 83–88 (1977). MR 454643, DOI 10.1090/S0002-9939-1977-0454643-7
- Scott McCullough and Stefan Richter, Bergman-type reproducing kernels, contractive divisors, and dilations, J. Funct. Anal. 190 (2002), no. 2, 447–480. MR 1899491, DOI 10.1006/jfan.2001.3874
- Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
- Nikolai K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and Toeplitz; Translated from the French by Andreas Hartmann. MR 1864396
- Stefan Richter, Invariant subspaces in Banach spaces of analytic functions, Trans. Amer. Math. Soc. 304 (1987), no. 2, 585–616. MR 911086, DOI 10.1090/S0002-9947-1987-0911086-8
- Marvin Rosenblum and James Rovnyak, Hardy classes and operator theory, Dover Publications, Inc., Mineola, NY, 1997. Corrected reprint of the 1985 original. MR 1435287
- L. A. Rubel and A. L. Shields, The second duals of certain spaces of analytic functions, J. Austral. Math. Soc. 11 (1970), 276–280. MR 0276744, DOI 10.1017/S1446788700006649
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- Kristian Seip, Beurling type density theorems in the unit disk, Invent. Math. 113 (1993), no. 1, 21–39. MR 1223222, DOI 10.1007/BF01244300
- Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. MR 0361899
- Zhijian Wu and Liming Yang, The codimension-$1$ property in Bergman spaces over planar regions, Michigan Math. J. 45 (1998), no. 2, 369–373. MR 1637674, DOI 10.1307/mmj/1030132188
- Li Ming Yang, Invariant subspaces of the Bergman space and some subnormal operators in $\mathbf A_1\sbs \mathbf A_2$, Michigan Math. J. 42 (1995), no. 2, 301–310. MR 1342492, DOI 10.1307/mmj/1029005230
Additional Information
- Alexandru Aleman
- Affiliation: Department of Mathematics, Lund University, P.O. Box 118, S-221 00 Lund, Sweden
- Email: Aleman@maths.lth.se
- Stefan Richter
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
- MR Author ID: 215743
- Email: Richter@math.utk.edu
- Carl Sundberg
- Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
- Email: Sundberg@math.utk.edu
- Received by editor(s): July 11, 2005
- Published electronically: February 12, 2007
- Additional Notes: Part of this work was done while the second author visited Lund University. He would like to thank the Mathematics Department for its hospitality. Furthermore, work of the first author was supported by the Royal Swedish Academy of Sciences and work of the second and third authors was supported by the U. S. National Science Foundation.
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 359 (2007), 3369-3407
- MSC (2000): Primary 47B32, 46E22; Secondary 30H05, 46E20
- DOI: https://doi.org/10.1090/S0002-9947-07-04258-4
- MathSciNet review: 2299460