Norm of the Hilbert matrix on Bergman and Hardy spaces and a theorem of Nehari type

The Hilbert matrix induces a bounded operator on most Hardy and Bergman spaces, as was shown by Diamantopoulos and Siskakis. We generalize this for any Hankel operator on Hardy spaces by using a result of Hollenbeck and Verbitsky on the Riesz projection and also compute the exact value of the norm of the Hilbert matrix. Using a new technique, we determine the norm of the Hilbert matrix on a wide range of Bergman spaces.