Vol. 53, No. 1, 1974

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Spectral distribution of the sum of self-adjoint operators

Arthur Larry Lieberman

Vol. 53 (1974), No. 1, 211–216

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 PK[λ,) be the spectral projection of K for the interval [λ,) and let ΛK(x) = sup{λ|m(PK[λ,)) x}. Then ΛK+L(x + r) ΛK(x)+ ΛL(r). If K L, then ΛK(x) ΛL(x). If L is bounded, then ΛLKL(x) L2ΛK(x) for x m(PK[0,)). If q = m (support (L)) and q < , then ΛK(x + q) ΛK+L(x); if μ = Λ|K|(q), then K + Lp KPK(μ,μ)p for 1 p .

Mathematical Subject Classification 2000
Primary: 46L10
Secondary: 47B25
Received: 12 February 1973
Published: 1 July 1974
Arthur Larry Lieberman