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Nontangential limits in Pt(µ)-spaces and the index of invariant subgroups

Aleman, Alexandru LU ; Richter, Stefan and Sundberg, Carl (2009) In Annals of Mathematics 169(2). p.449-490
Abstract
Abstract















Let μ be a finite positive

measure on the closed disk D¯

in the complex plane, let 1 ≤ t < ∞,

and let Pt(μ)

denote the closure of the analytic polynomials in

Lt(μ). We suppose

that D

is the set of analytic bounded point evaluations for

Pt(μ), and

that Pt(μ)

contains no nontrivial characteristic functions. It is then known that the restriction of

μ to

∂D must be of the form

h|dz|. We prove that every

function f ∈ Pt(μ) has nontangential

limits at h|dz|-almost

every point of ∂D,

and the... (More)
Abstract















Let μ be a finite positive

measure on the closed disk D¯

in the complex plane, let 1 ≤ t < ∞,

and let Pt(μ)

denote the closure of the analytic polynomials in

Lt(μ). We suppose

that D

is the set of analytic bounded point evaluations for

Pt(μ), and

that Pt(μ)

contains no nontrivial characteristic functions. It is then known that the restriction of

μ to

∂D must be of the form

h|dz|. We prove that every

function f ∈ Pt(μ) has nontangential

limits at h|dz|-almost

every point of ∂D,

and the resulting boundary function agrees with

f as an

element of Lt(h|dz|).



Our proof combines methods from James E. Thomson’s proof of the existence of bounded point

evaluations for Pt(μ)

whenever Pt(μ)≠Lt(μ)

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 ν

in the complex plane we are able to describe locations of bounded point evaluations

for Pt(ν) 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 < ∞ dim

ℳ∕zℳ = 1 for every nonzero

invariant subspace ℳ

of Pt(μ) if and

only if h≠0.



We also investigate the boundary behaviour of the functions in

Pt(μ) near the

points z ∈ ∂D

where h(z) = 0. In

particular, for 1 < t < ∞

we show that there are interpolating sequences for

Pt(μ)

that accumulate nontangentially almost everywhere on

{z : h(z) = 0}. (Less)
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type
Contribution to journal
publication status
published
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in
Annals of Mathematics
volume
169
issue
2
pages
449 - 490
publisher
Annals of Mathematics
ISSN
0003-486X
language
English
LU publication?
yes
id
bbe3a04b-1640-4e05-a55e-9333d23f70a8 (old id 1467113)
date added to LUP
2009-09-07 14:44:26
date last changed
2016-04-15 19:56:24
@article{bbe3a04b-1640-4e05-a55e-9333d23f70a8,
  abstract     = {Abstract<br/><br>
<br/><br>
<br/><br>
	 	<br/><br>
		 <br/><br>
			<br/><br>
				<br/><br>
<br/><br>
Let μ be a finite positive<br/><br>
measure on the closed disk D¯<br/><br>
in the complex plane, let 1 ≤ t &lt; ∞,<br/><br>
and let Pt(μ)<br/><br>
denote the closure of the analytic polynomials in<br/><br>
Lt(μ). We suppose<br/><br>
that D<br/><br>
is the set of analytic bounded point evaluations for<br/><br>
Pt(μ), and<br/><br>
that Pt(μ)<br/><br>
contains no nontrivial characteristic functions. It is then known that the restriction of<br/><br>
μ to<br/><br>
∂D must be of the form<br/><br>
h|dz|. We prove that every<br/><br>
function f ∈ Pt(μ) has nontangential<br/><br>
limits at h|dz|-almost<br/><br>
every point of ∂D,<br/><br>
and the resulting boundary function agrees with<br/><br>
f as an<br/><br>
element of Lt(h|dz|).<br/><br>
<br/><br>
 Our proof combines methods from James E. Thomson’s proof of the existence of bounded point<br/><br>
evaluations for Pt(μ)<br/><br>
whenever Pt(μ)≠Lt(μ)<br/><br>
with Xavier Tolsa’s remarkable recent results on analytic capacity. These methods allow<br/><br>
us to refine Thomson’s results somewhat. In fact, for a general compactly supported<br/><br>
measure ν<br/><br>
in the complex plane we are able to describe locations of bounded point evaluations<br/><br>
for Pt(ν) in<br/><br>
terms of the Cauchy transform of an annihilating measure.<br/><br>
<br/><br>
 As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We<br/><br>
show that for 1 &lt; t &lt; ∞ dim<br/><br>
ℳ∕zℳ = 1 for every nonzero<br/><br>
invariant subspace ℳ<br/><br>
of Pt(μ) if and<br/><br>
only if h≠0.<br/><br>
<br/><br>
 We also investigate the boundary behaviour of the functions in<br/><br>
Pt(μ) near the<br/><br>
points z ∈ ∂D<br/><br>
where h(z) = 0. In<br/><br>
particular, for 1 &lt; t &lt; ∞<br/><br>
we show that there are interpolating sequences for<br/><br>
Pt(μ)<br/><br>
that accumulate nontangentially almost everywhere on<br/><br>
{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 Pt(µ)-spaces and the index of invariant subgroups},
  volume       = {169},
  year         = {2009},
}