On the Hilbert matrix. I

Abstract

where the best possible constant Mk is 7r for k ? 1/2 and 7r csc 7rk I for 1/2 <k. Thus, when considered as a linear operator on the complex sequential Hilbert space 12, Hk is a bounded symmetric operator. Magnus [8] showed that the 12 spectrum of Ho is purely continuous and consists of the interval [0, 7r]. In this note we shall exhibit for each real k a monotone function Pk(X) and an isometric map Vk of '2 onto L2(dpk) such that VkHkV17' is a multiplication operator. This will allow us to determine the spectral nature of Hk. In [9] we studied an isomorphism of 12 with L2(0, oo) that transforms the Hilbert operator Hk into an integral operator which we shall now denote by 30k,1/2. It can be easily checked that 0k,1/2 formally commutes with the differential operator Lk which is defined below. Indeed, we shall prove that 30k,1/2=ir sech 7rL1/2. Since Lk can be diagonalized by a now standard procedure so 32k,1/2 and hence Hk can be diagonalized.