Nontangential limits in P-t(mu)-spaces and the index of invariant subspaces
(2009) In Annals of Mathematics 169(2). p.449-490- Abstract
- Let mu be a finite positive measure on the closed disk (D) over bar in the complex plane, let 1 <= t < infinity, and let P-t(mu) denote the closure of the analytic polynomials in L-t(mu). We suppose that D is the set of analytic bounded point evaluations for P-t(mu), and that P-t(mu) contains no nontrivial characteristic functions. It is then known that the restriction of mu to partial derivative D must be of the form h vertical bar dz vertical bar. We prove that every function f is an element of P-t(mu) has nontangential limits at h vertical bar dz vertical bar-almost every point of partial derivative D, and the resulting boundary function agrees with f as an element of L-t(h vertical bar dz vertical bar). Our proof combines methods... (More)
- Let mu be a finite positive measure on the closed disk (D) over bar in the complex plane, let 1 <= t < infinity, and let P-t(mu) denote the closure of the analytic polynomials in L-t(mu). We suppose that D is the set of analytic bounded point evaluations for P-t(mu), and that P-t(mu) contains no nontrivial characteristic functions. It is then known that the restriction of mu to partial derivative D must be of the form h vertical bar dz vertical bar. We prove that every function f is an element of P-t(mu) has nontangential limits at h vertical bar dz vertical bar-almost every point of partial derivative D, and the resulting boundary function agrees with f as an element of L-t(h vertical bar dz vertical bar). Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for P-t(mu) whenever P-t(mu) not equal L-t(mu) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure nu in the complex plane we are able to describe locations of bounded point evaluations for P-t(nu) in terms of the Cauchy transform of an annihilating measure. As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < infinity dim M/zM = 1 for every nonzero invariant subspace M of P-t(mu) if and only if h not equal 0. We also investigate the boundary behaviour of the functions in P-t(mu) near the points z is an element of partial derivative D where h(z) = 0. In particular, for 1 < t < infinity we show that there are interpolating sequences for P-t(mu) that accumulate nontangentially almost everywhere on {z : h(z) = 0}. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1476823
- author
- Aleman, Alexandru LU ; Richter, Stefan and Sundberg, Carl
- organization
- publishing date
- 2009
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Annals of Mathematics
- volume
- 169
- issue
- 2
- pages
- 449 - 490
- publisher
- Annals of Mathematics
- external identifiers
-
- wos:000269339200002
- ISSN
- 0003-486X
- language
- English
- LU publication?
- yes
- id
- f7f4e811-d85d-4c79-9d34-76a99e708285 (old id 1476823)
- alternative location
- http://annals.princeton.edu/annals/2009/169-2/p02.xhtml
- date added to LUP
- 2016-04-01 14:50:13
- date last changed
- 2021-09-27 04:51:38
@article{f7f4e811-d85d-4c79-9d34-76a99e708285,
abstract = {{Let mu be a finite positive measure on the closed disk (D) over bar in the complex plane, let 1 <= t < infinity, and let P-t(mu) denote the closure of the analytic polynomials in L-t(mu). We suppose that D is the set of analytic bounded point evaluations for P-t(mu), and that P-t(mu) contains no nontrivial characteristic functions. It is then known that the restriction of mu to partial derivative D must be of the form h vertical bar dz vertical bar. We prove that every function f is an element of P-t(mu) has nontangential limits at h vertical bar dz vertical bar-almost every point of partial derivative D, and the resulting boundary function agrees with f as an element of L-t(h vertical bar dz vertical bar). Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for P-t(mu) whenever P-t(mu) not equal L-t(mu) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure nu in the complex plane we are able to describe locations of bounded point evaluations for P-t(nu) in terms of the Cauchy transform of an annihilating measure. As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < infinity dim M/zM = 1 for every nonzero invariant subspace M of P-t(mu) if and only if h not equal 0. We also investigate the boundary behaviour of the functions in P-t(mu) near the points z is an element of partial derivative D where h(z) = 0. In particular, for 1 < t < infinity we show that there are interpolating sequences for P-t(mu) that accumulate nontangentially almost everywhere on {z : h(z) = 0}.}},
author = {{Aleman, Alexandru and Richter, Stefan and Sundberg, Carl}},
issn = {{0003-486X}},
language = {{eng}},
number = {{2}},
pages = {{449--490}},
publisher = {{Annals of Mathematics}},
series = {{Annals of Mathematics}},
title = {{Nontangential limits in P-t(mu)-spaces and the index of invariant subspaces}},
url = {{http://annals.princeton.edu/annals/2009/169-2/p02.xhtml}},
volume = {{169}},
year = {{2009}},
}