Geodesic
Hilbert inequality for vector valued functions
Archivum mathematicum, Tome 47 (2011) no. 3, pp. 229-243
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $.
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 $.
Classification : 26D15
Keywords: Hardy-Hilbert’s integral inequality; $\beta $-function; Hölder’s inequality 
@article{ARM_2011_47_3_a4,
     author = {Das, Namita and Sahoo, Srinibas},
     title = {Hilbert inequality for vector valued functions},
     journal = {Archivum mathematicum},
     pages = {229--243},
     year = {2011},
     volume = {47},
     number = {3},
     mrnumber = {2852383},
     zbl = {1249.26033},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2011_47_3_a4/}
}
TY  - JOUR
AU  - Das, Namita
AU  - Sahoo, Srinibas
TI  - Hilbert inequality for vector valued functions
JO  - Archivum mathematicum
PY  - 2011
SP  - 229
EP  - 243
VL  - 47
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/ARM_2011_47_3_a4/
LA  - en
ID  - ARM_2011_47_3_a4
ER  - 
%0 Journal Article
%A Das, Namita
%A Sahoo, Srinibas
%T Hilbert inequality for vector valued functions
%J Archivum mathematicum
%D 2011
%P 229-243
%V 47
%N 3
%U http://geodesic.mathdoc.fr/item/ARM_2011_47_3_a4/
%G en
%F ARM_2011_47_3_a4
Das, Namita; Sahoo, Srinibas. Hilbert inequality for vector valued functions. Archivum mathematicum, Tome 47 (2011) no. 3, pp. 229-243. http://geodesic.mathdoc.fr/item/ARM_2011_47_3_a4/

[1] Bercovici, H.: Operator theory and arithmetic in $H^{\infty }$. no. 26, Math. Surveys Monogr., 1988. | MR

[2] Duren, P. L.: Theory of $\mathcal{H}^p$ Spaces. Academic Press, New York, 1970. | MR

[3] Hardy, G. H.: Note on a theorem of Hilbert concerning series of positive terms. Proc. London Math. Soc. 23 (2) (1925), 45–46.

[4] Hardy, G. H., Littlewood, J. E., Polya, G.: Inequalities. Cambridge University Press, Cambridge, 1952. | MR | Zbl

[5] Kostrykin, V., Makarov, K. A.: On Krein’s example. arXiv:math/0606249v1 [math.SP], 10 June 2006. | MR | Zbl

[6] Mintrinovic, D. S., Pecaric, J. E., Fink, A. M.: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Boston, 1991. | MR

[7] Partington, J. R.: An introduction to Hankel operator. vol. 13, London Math. Soc. Stud. Texts, 1988. | MR

[8] Power, S. C.: Hankel operators on Hilbert space. Bull. London Math. Soc. 12 (1980), 422–442. | DOI | MR | Zbl