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Free hypercontractivity. (English) Zbl 0874.46049
Summary: 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 $$q\in[-1,1]$$, interpolating between the bosonic and the fermionic functors which correspond to $$q=+1$$ and $$q=-1$$ respectively.

##### MSC:
 46N50 Applications of functional analysis in quantum physics 81S10 Geometry and quantization, symplectic methods
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