A Remark on Hypercontractive Semigroups and Operator Ideals

In this note, we answer a question raised by Johnson and Schechtman [7], about the hypercontractive semigroup on {−1, 1}. More generally, we prove the theorem below. 2000 Math. Subject Classification: Primary 47D20. Seconday 47L20. In this note, we answer a question raised by Johnson and Schechtman [7]. We refer to their paper for motivation and background. In addition, we refer the reader to [1, 2, 5, 8] for examples of hypercontractive semigroups and related background (e.g. the classical works of Bonami, Nelson, Federbush, Gross, Beckner,...). Theorem 1. Let 1 < p < 2. Let (T (t))t>0 be a holomorphic semigroup on Lp (relative to a probability space). Assume the following mild form of hypercontractivity: for some large enough number s > 0, T (s) is bounded from Lp to L2. Then for any t > 0, T (t) is in the norm closure in B(Lp) (denoted by Γ2) of the subset (denoted by Γ2) formed by the operators mapping Lp to L2 (a fortiori these operators factor through a Hilbert space). Proof. Unless specified otherwise, by the norm of T (t) we mean its norm as acting from Lp to itself. We will consider the compact set K such that its boundary is a triangle with vertices 0, a ± ib, where a, b > 0 are chosen (by holomorphy of T ) so that t 7→ T (t) is bounded on K. By the semigroup property, t 7→ T (t) is also bounded on s +K. Moreover, for any t ∈ K, the factorization T (s+ t) = T (t)T (s) shows that T (s+ t) ∈ Γ2. More precisely, by assumption we have constants c, d such that ‖T (s) : Lp → L2‖ = c < ∞ and sup t∈K ‖T (t) : Lp → Lp‖ = d < ∞. ∗Partially supported by NSF grant No. 0503688