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An integral operator on $H\sp p$

Aleman, Alexandru LU and Siskakis, Aristomenis G (1995) In Complex Variables, Theory & Application 28(2). p.149-158
Abstract
To every analytic function $g$ in the unit disk $\bold D$ one associates the integral operator $T_g(f)(z)\coloneq z^{-1}\int^z_0f(\xi)g'(\xi)d\xi$ on spaces of analytic functions in $\bold D$. Operators of this form appear naturally in complex analysis. For instance, the choice $g(z)\equiv z$ leads to the integration operator, and the choice $g(z)=\log(1/(1-z))$ leads to the Cesàro operator.



The notation used in the paper is standard. For $0<p\leq\infty,\ H^p$ denotes the Hardy space, $B_p$ denotes the analytic Besov-$p$ space, and $S_p(H^2)$ denotes the Schatten-$p$ class of operators on $H^2$. BMOA and VMOA denote the spaces of analytic functions of bounded [respectively, vanishing] mean oscillation.

... (More)
To every analytic function $g$ in the unit disk $\bold D$ one associates the integral operator $T_g(f)(z)\coloneq z^{-1}\int^z_0f(\xi)g'(\xi)d\xi$ on spaces of analytic functions in $\bold D$. Operators of this form appear naturally in complex analysis. For instance, the choice $g(z)\equiv z$ leads to the integration operator, and the choice $g(z)=\log(1/(1-z))$ leads to the Cesàro operator.



The notation used in the paper is standard. For $0<p\leq\infty,\ H^p$ denotes the Hardy space, $B_p$ denotes the analytic Besov-$p$ space, and $S_p(H^2)$ denotes the Schatten-$p$ class of operators on $H^2$. BMOA and VMOA denote the spaces of analytic functions of bounded [respectively, vanishing] mean oscillation.



The main results of the paper are as follows. Theorem 1: Let $g$ be an analytic function in $\bold D$ and let $1\leq p<\infty$. Then $T_g$ is bounded on $H^p$ if and only if $g\in{\rm BMOA}$. Theorem 2: Let $g$ be an analytic function in $\bold D$. (i) If $1<p<\infty$ then $T_g\in S_p(H^2)$ if and only if $g\in B_p$. (ii) If $0<p\leq 1$ then $T_g\in S_p(H^2)$ if and only if $g$ is constant. Theorem 1 generalizes an earlier result of Ch. Pommerenke [Comment. Math. Helv. 52 (1977), no. 4, 591--602; MR0454017 (56 #12268)] in the context of $H^2$. It also implies the following corollary: Let $g$ be an analytic function on $\bold D$ and let $1\leq p<\infty$. Then $T_g$ is compact on $H^p$ if and only if $g\in{\rm VMOA}$. The main tool in the proof of Theorem 2 is Luecking's results [D. H. Luecking, J. Funct. Anal. 73 (1987), no. 2, 345--368; MR0899655 (88m:47046)] on Cauchy transforms of Borel measures $\mu$ on $\bold D$. $$Q_\mu(f)(w)\coloneq\int_{\bf D}\frac{f(z)}{1-w\overline z}d\mu(z).$$ Let $\{R_j\}^\infty_{j=1}$ be disjoint ``Carleson cubes'' which cover $\bold D$, and let $l(R_j)$ be the distance from the center of $R_j$ to $\partial\bold D$. Then Luecking's theorem [op. cit.] says that $Q_µ\in S_p(H^2)$ if and only if $\sum^\infty_{j=1}(\mu(R_j)/l(R_j))^p<\infty$. The connection to Luecking's theorem is via the relation $T^\ast_gT_g=Q_\mu$ with $d\mu(z)=2|g'(z)|^2\log(1/|z|)dm(z)$, where $m$ is Lebesgue measure. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Complex Variables, Theory & Application
volume
28
issue
2
pages
149 - 158
publisher
New York ; Gordon and Breach, 1982-
ISSN
1563-5066
DOI
10.1080/17476939508814844
language
English
LU publication?
no
id
1ea4416d-da8c-488f-94d1-51a298c9169d (old id 1467247)
date added to LUP
2009-09-16 13:34:39
date last changed
2016-06-29 09:13:50
@article{1ea4416d-da8c-488f-94d1-51a298c9169d,
  abstract     = {To every analytic function $g$ in the unit disk $\bold D$ one associates the integral operator $T_g(f)(z)\coloneq z^{-1}\int^z_0f(\xi)g'(\xi)d\xi$ on spaces of analytic functions in $\bold D$. Operators of this form appear naturally in complex analysis. For instance, the choice $g(z)\equiv z$ leads to the integration operator, and the choice $g(z)=\log(1/(1-z))$ leads to the Cesàro operator. <br/><br>
<br/><br>
The notation used in the paper is standard. For $0&lt;p\leq\infty,\ H^p$ denotes the Hardy space, $B_p$ denotes the analytic Besov-$p$ space, and $S_p(H^2)$ denotes the Schatten-$p$ class of operators on $H^2$. BMOA and VMOA denote the spaces of analytic functions of bounded [respectively, vanishing] mean oscillation. <br/><br>
<br/><br>
The main results of the paper are as follows. Theorem 1: Let $g$ be an analytic function in $\bold D$ and let $1\leq p&lt;\infty$. Then $T_g$ is bounded on $H^p$ if and only if $g\in{\rm BMOA}$. Theorem 2: Let $g$ be an analytic function in $\bold D$. (i) If $1&lt;p&lt;\infty$ then $T_g\in S_p(H^2)$ if and only if $g\in B_p$. (ii) If $0&lt;p\leq 1$ then $T_g\in S_p(H^2)$ if and only if $g$ is constant. Theorem 1 generalizes an earlier result of Ch. Pommerenke [Comment. Math. Helv. 52 (1977), no. 4, 591--602; MR0454017 (56 #12268)] in the context of $H^2$. It also implies the following corollary: Let $g$ be an analytic function on $\bold D$ and let $1\leq p&lt;\infty$. Then $T_g$ is compact on $H^p$ if and only if $g\in{\rm VMOA}$. The main tool in the proof of Theorem 2 is Luecking's results [D. H. Luecking, J. Funct. Anal. 73 (1987), no. 2, 345--368; MR0899655 (88m:47046)] on Cauchy transforms of Borel measures $\mu$ on $\bold D$. $$Q_\mu(f)(w)\coloneq\int_{\bf D}\frac{f(z)}{1-w\overline z}d\mu(z).$$ Let $\{R_j\}^\infty_{j=1}$ be disjoint ``Carleson cubes'' which cover $\bold D$, and let $l(R_j)$ be the distance from the center of $R_j$ to $\partial\bold D$. Then Luecking's theorem [op. cit.] says that $Q_µ\in S_p(H^2)$ if and only if $\sum^\infty_{j=1}(\mu(R_j)/l(R_j))^p&lt;\infty$. The connection to Luecking's theorem is via the relation $T^\ast_gT_g=Q_\mu$ with $d\mu(z)=2|g'(z)|^2\log(1/|z|)dm(z)$, where $m$ is Lebesgue measure.},
  author       = {Aleman, Alexandru and Siskakis, Aristomenis G},
  issn         = {1563-5066},
  language     = {eng},
  number       = {2},
  pages        = {149--158},
  publisher    = {New York ; Gordon and Breach, 1982-},
  series       = {Complex Variables, Theory & Application},
  title        = {An integral operator on $H\sp p$},
  url          = {http://dx.doi.org/10.1080/17476939508814844},
  volume       = {28},
  year         = {1995},
}