OPTIMAL LOG-SOBOLEV INEQUALITY AND HYPERCONTRACTIVITY FOR POSITIVE SEMIGROUPS ON $M_2({\mathbb C})$

Abstract

We study positivity and contractivity properties for semigroups on , compute the optimal log-Sobolev constant and prove hypercontractivity for the class of positive semigroups leaving invariant both subspaces generated by the Pauli matrices σ₀, σ₃ and σ₁, σ₂. The optimal log-Sobolev constant turns out to be bigger than the usual one arising in several commutative and noncommutative contexts when the semigroup acts on the off-diagonal matrices faster than on diagonal matrices. These results are applied to the semigroup of the Wigner–Weisskopf atom.