Hilbert inequality for vector valued functions. (Q2897392)

The authors investigate a Hilbert type inequality and related topics for functions and sequences attaining values in a Hilbert space. One of the results of the paper reads as follows.NEWLINENEWLINETheorem: Let \(\{x_n\}\), \(\{y_n\}\) be two sequences of elements of a Hilbert space \(\mathcal {H}\) such that \(0<\sum _{n=1}^\infty \| x_n\| ^2<\infty \), \(0<\sum _{n=1}^\infty \| y_n\| ^2<\infty \). Then NEWLINE\[NEWLINE \sum _{m=1}^\infty \sum _{n=1}^\infty \frac {| \langle x_m,y_n\rangle | }{m+n}<\pi \left (\sum _{m=1}^{\infty } \| x_m\| ^2\right )^{1/2} \left (\sum _{n=1}^{\infty } \| y_n\| ^2\right )^{1/2}, NEWLINE\]NEWLINE where the constant \(\pi \) is the best possible. Various modifications of this statement are presented as well.