Positivity improving operators and hypercontractivity

Abstract

Here (1.1) is the famous Nelson hypercontractivity inequality [14], which has proved to be extremely useful in Euclidean quantum field theory (see e.g. [16]). Like (1.1) the estimate (1.2) follows very easily from the It6 calculus of Brownian motion [3]. Here, in order to make the paper self-contained and to introduce the reader to some general principles of interest, we submit a new proof of (1.2) by mimicing a standard proof of (1.1). We should like to add that (1.2) has at least a formal resemblance to the familiar correlation inequalities from statistical mechanics (see e.g. [11, 12, 16]). Using (1.1) and (1.2) it is simple to obtain similar estimates for the biquantization of a linear contraction on Hilbert space (Sect. 4). Now suppose A is a self-adjoint operator on L 2 of a probability space and set A(A)=maxspec(A)-sup(spec(A)\{maxspec(A)}), where spec(A) denotes the spectrum of A. Below we will state conditions that guarantee that the gap A (A) is proper. In fact, a simplified version of our main result (Theorem 5.3) gives a positive numerical lower bound of A (A) if (i) A is positivity preserving, (ii) sup { IlZf I1~; [I f [12 = 1 } = K 2, and (iii) inf{ [IZfllp/~v_~; I l f l l , = 1, f>0} = k > 0 some 0 < p < 1. Here (i)-(iii) imply that A has a unique ground state. It shall be empha-