On Finite and Infinite Decomposition of Some Hilbert’s Type Inequalities

In this work, some Hardy-Hilbert's integral inequalities with the best possible constants is proved. Also, some finite and infinite decompositions of sometype Hardy-Hilbert's integral operators is given. Indeed, for a non-negative kernel K, two Kernels K1 and K2 is given such that TK = TK1 + TK2 and ∥TK∥ =∥TK1∥ + ∥TK2∥. So, the space of bounded linear operators is strictly convex.Also, as an application of infinite decomposition of some Hardy-Hilbert's integraloperators, the convergence of some series of hypergeometric functions is given.