| Title:
|
Hilbert inequality for vector valued functions (English) |
| Author:
|
Das, Namita |
| Author:
|
Sahoo, Srinibas |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
47 |
| Issue:
|
3 |
| Year:
|
2011 |
| Pages:
|
229-243 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we consider a class of Hankel operators with operator valued symbols on the Hardy space ${ \mathcal{H}}_{\Xi }^2(\mathbb{T})$ where $\Xi $ is a separable infinite dimensional Hilbert space and showed that these operators are unitarily equivalent to a class of integral operators in $L^2(0, \infty )\otimes \Xi .$ We then obtained a generalization of Hilbert inequality for vector valued functions. In the continuous case the corresponding integral operator has matrix valued kernels and in the discrete case the sum involves inner product of vectors in the Hilbert space $\Xi $. (English) |
| Keyword:
|
Hardy-Hilbert’s integral inequality |
| Keyword:
|
$\beta $-function |
| Keyword:
|
Hölder’s inequality |
| MSC:
|
26D15 |
| idZBL:
|
Zbl 1249.26033 |
| idMR:
|
MR2852383 |
| . |
| Date available:
|
2011-11-11T08:53:35Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141709 |
| . |
| Reference:
|
[1] Bercovici, H.: Operator theory and arithmetic in $H^{\infty }$.no. 26, Math. Surveys Monogr., 1988. MR 0954383 |
| Reference:
|
[2] Duren, P. L.: Theory of $\mathcal{H}^p$ Spaces.Academic Press, New York, 1970. MR 0268655 |
| Reference:
|
[3] Hardy, G. H.: Note on a theorem of Hilbert concerning series of positive terms.Proc. London Math. Soc. 23 (2) (1925), 45–46. |
| Reference:
|
[4] Hardy, G. H., Littlewood, J. E., Polya, G.: Inequalities.Cambridge University Press, Cambridge, 1952. Zbl 0047.05302, MR 0046395 |
| Reference:
|
[5] Kostrykin, V., Makarov, K. A.: On Krein’s example.arXiv:math/0606249v1 [math.SP], 10 June 2006. Zbl 1143.47019, MR 2383512 |
| Reference:
|
[6] Mintrinovic, D. S., Pecaric, J. E., Fink, A. M.: Inequalities Involving Functions and Their Integrals and Derivatives.Kluwer Academic Publishers, Boston, 1991. MR 1190927 |
| Reference:
|
[7] Partington, J. R.: An introduction to Hankel operator.vol. 13, London Math. Soc. Stud. Texts, 1988. MR 0985586 |
| Reference:
|
[8] Power, S. C.: Hankel operators on Hilbert space.Bull. London Math. Soc. 12 (1980), 422–442. Zbl 0446.47015, MR 0593961, 10.1112/blms/12.6.422 |
| . |
