Jensen’s operator inequality for functions of two variables

Abstract

The operator convex functions of two variables are characterized in terms of a non-commutative generalization of Jensen's inequality. 1. FUNCTIONAL CALCULUS FOR FUNCTIONS OF SEVERAL VARIABLES The tensor product of two square matrices A = (aij) and B of order n can be represented as the matrix allB ... alnB (1) 1@B anl B .*.** aB / which is of order n2. However, if A and B are block matrices ( A21 A22) B = B21 B22) of order 2n, then it is often more convenient to represent the tensor product A 0 B as the block matrix / All 0 Bil All 0 B12 A12 0 Bil A12 0) B12 (2) A11? B21 A11 0 B22 A12 0 B21 A12 0 B22 (2) A21 B11 A21 0 B12 A22 X B1i1 A22 $' B12 A21 0 B21 A21 0 B22 A22 0 B21 A22 0 B22 / The definition according to (2) is unitarily equivalent to the definition according to (1), and no confusion will occur as long as the two representations are not mixed. The latter representation has the benefit of rendering formulas for block matrices more transparent and will be used throughout this paper. A similar representation will be used for tensor products of block matrices of bounded linear operators on a Hilbert space. Koranyi [10] considered functional calculus for functions of two variables. Let f: I x J -* R be a function of two variables defined on the product of two intervals, and let A, B be selfadjoint linear operators with finite spectra on a Hilbert space. If the spectrum of A is contained in I, and the spectrum of B is contained in J, Received by the editors February 2, 1996. 1991 Mathematics Subject Classification. Primary 47A63; Secondary 47A80, 47Bxx. ( 1997 American Mathematical Society 2093 This content downloaded from 207.46.13.176 on Mon, 20 Jun 2016 05:17:33 UTC All use subject to http://about.jstor.org/terms