An inequality for operators in a Hilbert space

Bertram Mond

Vol. 18 (1966), No. 1, 161–163

DOI: 10.2140/pjm.1966.18.161

Abstract

Let A be a self-adjoint operator on a Hilbert space H satisfying mI ≦ A ≦ MI, 0 < m < M. Set q = M∕m. Let j and k be real numbers, jk≠0, j < k. Then

for all x ∈ H(x≠0). Letting j = −1 and k = 1, this inequality reduces to (Ax,x)(A⁻¹x,x) ≦ [(M + m)²∕4mM](x,x)², the well-known Kantorovich Inequality.

Mathematical Subject Classification

Primary: 47.10

Secondary: 47.40

Milestones

Received: 11 March 1965

Published: 1 July 1966