Spectral distribution of the sum of self-adjoint operators

Arthur Larry Lieberman

Vol. 53 (1974), No. 1, 211–216

DOI: 10.2140/pjm.1974.53.211

Abstract

Using the techniques of noncommutative integration theory, classical results of Hermann Weyl concerning the positive eigenvalues of the sum of two self-adjoint compact operators are extended to self-adjoint operators which are measurable with respect to a gage space. Let (H,A,m) be a gage space and let K and L be self-adjoint operators which are measurable with respect to (H,A,m). Let P_K[λ,∞) be the spectral projection of K for the interval [λ,∞) and let Λ_K(x) = sup{λ|m(P_K[λ,∞)) ≧ x}. Then Λ_K+L(x + r) ≦ Λ_K(x)+ Λ_L(r). If K ≦ L, then Λ_K(x) ≦ Λ_L(x). If L is bounded, then Λ_LKL(x) ≦∥L∥²Λ_K(x) for x ≦ m(P_K[0,∞)). If q = m (support (L)) and q < ∞, then Λ_K(x + q) ≦ Λ_K+L(x); if μ = Λ_|K|(q), then ∥K + L∥_p ≧∥KP_K(−μ,μ)∥_p for 1 ≦ p ≦∞.

Mathematical Subject Classification 2000

Primary: 46L10

Secondary: 47B25

Milestones

Received: 12 February 1973

Published: 1 July 1974