Nontangential limits in Pt(mu)spaces and the index of invariant subspaces
(2009) In Annals of Mathematics 169(2). p.449490 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 Pt(mu) denote the closure of the analytic polynomials in Lt(mu). We suppose that D is the set of analytic bounded point evaluations for Pt(mu), and that Pt(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 Pt(mu) has nontangential limits at h vertical bar dz vertical baralmost every point of partial derivative D, and the resulting boundary function agrees with f as an element of Lt(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 Pt(mu) denote the closure of the analytic polynomials in Lt(mu). We suppose that D is the set of analytic bounded point evaluations for Pt(mu), and that Pt(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 Pt(mu) has nontangential limits at h vertical bar dz vertical baralmost every point of partial derivative D, and the resulting boundary function agrees with f as an element of Lt(h vertical bar dz vertical bar). Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for Pt(mu) whenever Pt(mu) not equal Lt(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 Pt(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 Pt(mu) if and only if h not equal 0. We also investigate the boundary behaviour of the functions in Pt(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 Pt(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
 0003486X
 language
 English
 LU publication?
 yes
 id
 f7f4e811d85d4c799d3476a99e708285 (old id 1476823)
 alternative location
 http://annals.princeton.edu/annals/2009/1692/p02.xhtml
 date added to LUP
 20160401 14:50:13
 date last changed
 20210927 04:51:38
@article{f7f4e811d85d4c799d3476a99e708285, 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 Pt(mu) denote the closure of the analytic polynomials in Lt(mu). We suppose that D is the set of analytic bounded point evaluations for Pt(mu), and that Pt(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 Pt(mu) has nontangential limits at h vertical bar dz vertical baralmost every point of partial derivative D, and the resulting boundary function agrees with f as an element of Lt(h vertical bar dz vertical bar). Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for Pt(mu) whenever Pt(mu) not equal Lt(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 Pt(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 Pt(mu) if and only if h not equal 0. We also investigate the boundary behaviour of the functions in Pt(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 Pt(mu) that accumulate nontangentially almost everywhere on {z : h(z) = 0}.}, author = {Aleman, Alexandru and Richter, Stefan and Sundberg, Carl}, issn = {0003486X}, language = {eng}, number = {2}, pages = {449490}, publisher = {Annals of Mathematics}, series = {Annals of Mathematics}, title = {Nontangential limits in Pt(mu)spaces and the index of invariant subspaces}, url = {http://annals.princeton.edu/annals/2009/1692/p02.xhtml}, volume = {169}, year = {2009}, }