Free Hypercontractivity

Abstract:

We prove the analog of Nelson's optimal hypercontractivity inequality, for the free second quantization (in the sense of Voiculescu) of a contraction between two real Hilbert spaces. The proof extends also to the case of the q-quantization functors of Bozejko, Kümmerer and Speicher, for \(\), interpolating between the bosonic and the fermionic functors which correspond to \(\) and \(\) respectively.