Analytic contractions, nontangential limits, and the index of invariant subspaces
Authors:
Alexandru Aleman, Stefan Richter and Carl Sundberg
Journal:
Trans. Amer. Math. Soc. 359 (2007), 33693407
MSC (2000):
Primary 47B32, 46E22; Secondary 30H05, 46E20
Published electronically:
February 12, 2007
MathSciNet review:
2299460
Fulltext PDF Free Access
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Similar Articles 
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Abstract: Let be a Hilbert space of analytic functions on the open unit disc such that the operator of multiplication with the identity function defines a contraction operator. In terms of the reproducing kernel for we will characterize the largest set such that for each , the meromorphic function has nontangential limits a.e. on . We will see that the question of whether or not has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of . We further associate with a second set , which is defined in terms of the norm on . For example, has the property that for all if and only if has linear Lebesgue measure 0. It turns out that a.e., by which we mean that has linear Lebesgue measure 0. We will study conditions that imply that a.e.. As one corollary to our results we will show that if dim and if there is a such that for all and all we have , then a.e. and the following four conditions are equivalent: (1) for some , (2) for all , , (3) has nonzero Lebesgue measure, (4) every nonzero invariant subspace of has index 1, i.e., satisfies dim .
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 [AR]
 Aleman, Alexandru, and Richter, Stefan, Some sufficient conditions for the division property of invariant subspaces in weighted Bergman spaces. J. Funct. Anal. 144 (1997), no. 2, 542556. MR 1432597 (98e:46030)
 [ARR]
 Aleman, Alexandru, Richter, Stefan, and Ross, William T., Pseudocontinuations and the backward shift. Indiana Univ. Math. J. 47 (1998), no. 1, 223276. MR 1631561 (2000i:47009)
 [ARS1]
 Aleman, Alexandru, Richter, Stefan, and Sundberg, Carl, The majorization function and the index of invariant subspaces in the Bergman spaces. J. Anal. Math. 86 (2002), 139182. MR 1894480 (2003g:30058)
 [ARS2]
 Aleman, Alexandru, Richter, Stefan, and Sundberg, Carl, Nontangential limits in spaces and the index of invariant subspaces, preprint.
 [ARS3]
 Aleman, Alexandru, Richter, Stefan, and Sundberg, Carl, 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., 404, American Mathematical Society, Providence, RI, 2006. MR 2244001
 [ABFP]
 Apostol, C., Bercovici, H., Foias, C., and Pearcy, C., Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. I. J. Funct. Anal. 63 (1985), no. 3, 369404. MR 0808268 (87i:47004a)
 [Ar]
 Aronszajn, N., Theory of reproducing kernels, Trans. Amer. Math. Soc. 68, (1950), 337404. MR 0051437 (14:479c)
 [Be]
 Bercovici, Hari, Factorization theorems and the structure of operators on Hilbert space. Ann. of Math. (2) 128 (1988), no. 2, 399413. MR 0960951 (89i:47032)
 [BSZ]
 Brown, Leon, Shields, Allen, and Zeller, Karl, On absolutely convergent exponential sums. Trans. Amer. Math. Soc. 96 (1960) 162183. MR 0142763 (26:332)
 [CEP]
 Chevreau, Bernard, Exner, George R., and Pearcy, Carl M., Boundary sets for a contraction. J. Operator Theory 34 (1995), no. 2, 347380. MR 1373328 (97b:47044)
 [Co]
 Conway, John B., A course in functional analysis. Second edition. Graduate Texts in Mathematics, 96. SpringerVerlag, New York, 1990. xvi+399 pp. ISBN: 0387972455 MR 1070713 (91e:46001)
 [Du]
 Duren, Peter L., Theory of spaces. Pure and Applied Mathematics, Vol. 38 Academic Press, New YorkLondon, 1970, xii+258 pp. MR 0268655 (42:3552)
 [DS]
 Duren, Peter, and Schuster, Alexander, Bergman spaces. Mathematical Surveys and Monographs, 100. American Mathematical Society, Providence, RI, 2004. x+318 pp. ISBN: 0821808109 MR 2033762 (2005c:30053)
 [Ga]
 Garnett, John B., Bounded analytic functions. Pure and Applied Mathematics, 96. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1981. xvi+467 pp. ISBN: 0122761502 MR 0628971 (83g:30037)
 [HKZ]
 Hedenmalm, Haakan, Korenblum, Boris, and Zhu, Kehe, Theory of Bergman spaces. Graduate Texts in Mathematics, 199. SpringerVerlag, New York, 2000. x+286 pp. ISBN: 0387987916 MR 1758653 (2001c:46043)
 [KT]
 Kriete, Thomas and Trent, Tavan, Growth near the boundary in spaces. Proc. Amer. Math. Soc. 62 (1976), no. 1, 8388 (1977). MR 0454643 (56:12892)
 [MR]
 McCullough, Scott and Richter, Stefan, Bergmantype reproducing kernels, contractive divisors, and dilations. J. Funct. Anal. 190 (2002), no. 2, 447480. MR 1899491 (2003c:47043)
 [NF]
 Sz.Nagy, B. and Foias, C., Harmonic analysis of operators on Hilbert space. Translated from the French and revised, NorthHolland Publishing Co., AmsterdamLondon; American Elsevier Publishing Co., Inc., New York. MR 0275190 (43:947)
 [Ni]
 Nikolski, Nikolai, Operators, Functions, and Systems: An Easy Reading, Volume 2: Model Operators and Systems, translated from the French by Andreas Hartmann and revised by the author. Mathematical Surveys and Monographs, 93. American Mathematical Society, Providence, RI, 2002. MR 1892647 (2003i:47001b)
 [Ri]
 Richter, Stefan, Invariant subspaces in Banach spaces of analytic functions. Trans. Amer. Math. Soc. 304 (1987), no. 2, 585616. MR 0911086 (88m:47056)
 [RR]
 Rosenblum, Marvin and Rovnyak, James, Hardy classes and operator theory. Corrected reprint of the 1985 original. Dover Publications, Inc., Mineola, NY, 1997. xiv+161 pp. ISBN: 0486695360 MR 1435287 (97j:47002)
 [RS]
 Rubel, L.A. and Shields, A.L., The second duals of certain spaces of analytic functions. J. Australian Math. Soc., XI, part 3, (1970), 276280. MR 0276744 (43:2484)
 [Ru]
 Rudin, Walter, Real and Complex Analysis, 3rd edition, McGrawHill, New York, 1987. MR 0924157 (88k:00002)
 [Se]
 Seip, Kristian, Beurling type density theorems in the unit disk. Invent. Math. 113 (1993), no. 1, 2139. MR 1223222 (94g:30033)
 [Sh]
 Shields, Allen L., Weighted shift operators and analytic function theory. Topics in operator theory, pp. 49128. Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974. MR 0361899 (50:14341)
 [WY]
 Wu, Zhijian and Yang, Liming, The codimension property in Bergman spaces over planar regions. Michigan Math. J. 45 (1998), no. 2, 369373. MR 1637674 (99g:46025)
 [Ya]
 Yang, Li Ming, Invariant subspaces of the Bergman space and some subnormal operators in . Michigan Math. J. 42 (1995), no. 2, 301310. MR 1342492 (96f:47013)
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Additional Information
Alexandru Aleman
Affiliation:
Department of Mathematics, Lund University, P.O. Box 118, S221 00 Lund, Sweden
Email:
Aleman@maths.lth.se
Stefan Richter
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 379961300
Email:
Richter@math.utk.edu
Carl Sundberg
Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 379961300
Email:
Sundberg@math.utk.edu
DOI:
http://dx.doi.org/10.1090/S0002994707042584
PII:
S 00029947(07)042584
Keywords:
Hilbert space of analytic functions,
contraction,
nontangential limits,
invariant subspaces,
index
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.
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
