Hilbert Inequality for Vector Valued Functions

Namita Das and Srinibas Sahoo

P.G. Department of Mathematics, Utkal University, Bhubaneswar, Orissa, India 751004
Department of Mathematics, Jatni College, Jatni, Khurda, Orissa, India 752050


Abstract: 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 $.

AMSclassification: primary 26D15.

Keywords: Hardy-Hilbert’s integral inequality, \beta -function, Hölder’s inequality.