# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Generalized integration operators on Hardy spaces

(2017) In Master's Theses in Mathematical Sciences MATM01 20171
Mathematics (Faculty of Sciences)
Abstract
Inspired by the study of generalized Cesaro operator T_g introduced by Aleman and Siskakis we study a variation of this operator,namely P(g,a) , depending on an analytic symbol g and an n - tuple of complex numbers, a . Regarding the boundedness properties of this operator we prove that P(g,a) is a bounded linear operator from H^p to itself if and only if g is an analytic function of bounded mean oscillation and compact if and only if g is of vanishing mean oscillation. Furthermore in the special case n=2, a=0 we completely characterized the functions g for which P(g,a) is bounded from H^p to H^q, 0<p,q. As an application of our theorem we prove a factorization theorem for any derivative of an \$ H^p \$ function, and also a theorem about... (More)
Inspired by the study of generalized Cesaro operator T_g introduced by Aleman and Siskakis we study a variation of this operator,namely P(g,a) , depending on an analytic symbol g and an n - tuple of complex numbers, a . Regarding the boundedness properties of this operator we prove that P(g,a) is a bounded linear operator from H^p to itself if and only if g is an analytic function of bounded mean oscillation and compact if and only if g is of vanishing mean oscillation. Furthermore in the special case n=2, a=0 we completely characterized the functions g for which P(g,a) is bounded from H^p to H^q, 0<p,q. As an application of our theorem we prove a factorization theorem for any derivative of an \$ H^p \$ function, and also a theorem about solutions of complex linear differential equations. (Less)
Popular Abstract (Swedish)
Inom komplex analys och operatorteori studerar man vanligtvis begränsade linjära operatorer mellan Banachrum bestående av analytiska funktioner. Detta görs för att kunna erhålla information om själva Banachrummets struktur.

Ett klassiskt exempel är Cesaros medelvärdes operator på H^p , ett Hardyrum bestående av analytiska funktioner. En generalisering av denna operator är det så kallade Cesaros generaliserad operator, Tg , som kan spåras tillbaka till arbetet av Ch. Pommerenke, 1970.
Operatorns egenskaper har varit ett aktivt forskningsområde i de senaste 20 åren. I detta arbete, som är inspirerad av studien av Tg , försöker vi ge svar till några frågor angående operatorns variation.
author
supervisor
organization
course
MATM01 20171
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Cesaro operator, Hardy spaces, Integration operators, Operator theory, Complex analysis
publication/series
Master's Theses in Mathematical Sciences
report number
LUNFMA-3094-2017
ISSN
1404-6342
other publication id
2017:E57
language
English
id
8924337
2017-10-03 15:21:11
date last changed
2017-10-03 15:21:11
```@misc{8924337,
abstract     = {Inspired by the study of generalized Cesaro operator T_g introduced by Aleman and Siskakis we study a variation of this operator,namely P(g,a) , depending on an analytic symbol g and an n - tuple of complex numbers, a . Regarding the boundedness properties of this operator we prove that P(g,a) is a bounded linear operator from H^p to itself if and only if g is an analytic function of bounded mean oscillation and compact if and only if g is of vanishing mean oscillation. Furthermore in the special case n=2, a=0 we completely characterized the functions g for which P(g,a) is bounded from H^p to H^q, 0<p,q. As an application of our theorem we prove a factorization theorem for any derivative of an \$ H^p \$ function, and also a theorem about solutions of complex linear differential equations.},
author       = {Chalmoukis, Nikolaos},
issn         = {1404-6342},
keyword      = {Cesaro operator,Hardy spaces,Integration operators,Operator theory,Complex analysis},
language     = {eng},
note         = {Student Paper},
series       = {Master's Theses in Mathematical Sciences},
title        = {Generalized integration operators on Hardy spaces},
year         = {2017},
}

```