On the Hilbert Matrix, Ii1

where the best possible constant Mk is it for k ^ 1/2 and ir\ csc irk\ for 1/2 <k. Thus, when considered as a linear operator on the complex sequential Hilbert space l2, Hk is a bounded symmetric operator. Magnus [8] showed that the l2 spectrum of H0 is purely continuous and consists of the interval [0, it]. In this note we shall exhibit for each real k a monotone function pk(\) and an isometric map Vk of Z2 onto L2(dpk) such that VkHkVkl is a multiplication operator. This will allow us to determine the spectral nature of IIk. In [9] we studied an isomorphism of I2 with L2(0, 00) that transforms the Hilbert operator Hk into an integral operator which we shall now denote by 3C*,i/2It can be easily checked that SCk,i/2 formally commutes with the differential operator Lk which is defined below. Indeed, we shall prove that 3Ck,i/2 = w sech irLk . Since Lk can be diagonalized by a now standard procedure so Xi,i/2 and hence Hk can be diagonalized.