The heat operator in infinite dimensions

Let (H,B) be an abstract Wiener space and let μs be the Gaussian measure on B with variance s. Let ∆ be the Laplacian (not the number operator), that is, a sum of squares of derivatives associated to an orthonormal basis of H. I will show that the heat operator exp(t∆/2) is a contraction operator from L2(B,μs) to L2(B,μs−t), for all t < s. More generally, the heat operator is a contraction from L(B,μs) to L (B,μs−t) for t < s, provided that p and q satisfy p− 1 q − 1 ≤ s s− t . I give two proofs of this result, both very elementary.