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), 3369-3407

MSC (2000):
Primary 47B32, 46E22; Secondary 30H05, 46E20

DOI:
https://doi.org/10.1090/S0002-9947-07-04258-4

Published electronically:
February 12, 2007

MathSciNet review:
2299460

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 .

**[AR]**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**, https://doi.org/10.1006/jfan.1996.2998**[ARR]**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**, https://doi.org/10.1512/iumj.1998.47.1583**[ARS1]**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**, https://doi.org/10.1007/BF02786647**[ARS2]**Aleman, Alexandru, Richter, Stefan, and Sundberg, Carl,*Nontangential limits in -spaces and the index of invariant subspaces*, preprint.**[ARS3]**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**, https://doi.org/10.1090/conm/404/07631**[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, 369-404. MR**0808268 (87i:47004a)****[Ar]**N. Aronszajn,*Theory of reproducing kernels*, Trans. Amer. Math. Soc.**68**(1950), 337–404. MR**0051437**, https://doi.org/10.1090/S0002-9947-1950-0051437-7**[Be]**Bercovici, Hari,*Factorization theorems and the structure of operators on Hilbert space.*Ann. of Math. (2) 128 (1988), no. 2, 399-413. MR**0960951 (89i:47032)****[BSZ]**Leon Brown, Allen Shields, and Karl Zeller,*On absolutely convergent exponential sums*, Trans. Amer. Math. Soc.**96**(1960), 162–183. MR**0142763**, https://doi.org/10.1090/S0002-9947-1960-0142763-8**[CEP]**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****[Co]**John B. Conway,*A course in functional analysis*, 2nd ed., Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1990. MR**1070713****[Du]**Peter L. Duren,*Theory of 𝐻^{𝑝} spaces*, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR**0268655****[DS]**Peter Duren and Alexander Schuster,*Bergman spaces*, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. MR**2033762****[Ga]**Garnett, John B.,*Bounded analytic functions.*Pure and Applied Mathematics, 96. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. xvi+467 pp. ISBN: 0-12-276150-2 MR**0628971 (83g:30037)****[HKZ]**Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu,*Theory of Bergman spaces*, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR**1758653****[KT]**Thomas Kriete and Tavan Trent,*Growth near the boundary in 𝐻²(𝜇) spaces*, Proc. Amer. Math. Soc.**62**(1976), no. 1, 83–88 (1977). MR**0454643**, https://doi.org/10.1090/S0002-9939-1977-0454643-7**[MR]**Scott McCullough and Stefan Richter,*Bergman-type reproducing kernels, contractive divisors, and dilations*, J. Funct. Anal.**190**(2002), no. 2, 447–480. MR**1899491**, https://doi.org/10.1006/jfan.2001.3874**[NF]**Béla Sz.-Nagy and Ciprian Foiaş,*Harmonic analysis of operators on Hilbert space*, Translated from the French and revised, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. MR**0275190****[Ni]**Nikolai K. Nikolski,*Operators, functions, and systems: an easy reading. Vol. 2*, Mathematical Surveys and Monographs, vol. 93, American Mathematical Society, Providence, RI, 2002. Model operators and systems; Translated from the French by Andreas Hartmann and revised by the author. MR**1892647****[Ri]**Richter, Stefan,*Invariant subspaces in Banach spaces of analytic functions.*Trans. Amer. Math. Soc. 304 (1987), no. 2, 585-616. MR**0911086 (88m:47056)****[RR]**Marvin Rosenblum and James Rovnyak,*Hardy classes and operator theory*, Dover Publications, Inc., Mineola, NY, 1997. Corrected reprint of the 1985 original. MR**1435287****[RS]**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****[Ru]**Rudin, Walter,*Real and Complex Analysis*, 3rd edition, McGraw-Hill, New York, 1987. MR**0924157 (88k:00002)****[Se]**Kristian Seip,*Beurling type density theorems in the unit disk*, Invent. Math.**113**(1993), no. 1, 21–39. MR**1223222**, https://doi.org/10.1007/BF01244300**[Sh]**Allen L. Shields,*Weighted shift operators and analytic function theory*, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math. Surveys, No. 13. MR**0361899****[WY]**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**, https://doi.org/10.1307/mmj/1030132188**[Ya]**Li Ming Yang,*Invariant subspaces of the Bergman space and some subnormal operators in 𝐀₁_s𝐀₂*, Michigan Math. J.**42**(1995), no. 2, 301–310. MR**1342492**, https://doi.org/10.1307/mmj/1029005230

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
47B32,
46E22,
30H05,
46E20

Retrieve articles in all journals with MSC (2000): 47B32, 46E22, 30H05, 46E20

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

Email:
Richter@math.utk.edu

**Carl Sundberg**

Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300

Email:
Sundberg@math.utk.edu

DOI:
https://doi.org/10.1090/S0002-9947-07-04258-4

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.