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Hardy-Hilbert’s integral inequality; $\beta $-function; Hölder’s inequality
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 $.
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