On Weakly Formulated Sylvester Equations and Applications

We use a “weakly formulated” Sylvester equation H1/2TM−1/2 −H−1/2TM1/2 = F to obtain new bounds for the rotation of spectral subspaces of a nonnegative selfadjoint operator in a Hilbert space. Our bound extends the known results of Davis and Kahan. Another application is a bound for the square root of a positive selfadjoint operator which extends the known rule: “The relative error in the square root is bounded by the one half of the relative error in the radicand”. Both bounds are illustrated on differential operators which are defined via quadratic forms. 1. Preliminaries In this work we will study properties of nonnegative selfadjoint operators in a Hilbert space which are close in the sense of the inequality (1.1) |h(φ, ψ)−m(φ, ψ)| ≤ η √ h[φ]m[ψ] where the sesquilinear forms h,m belong to the operators H,M respectively and m[ψ] = m(ψ, ψ), h[φ] = h(φ, φ). ByQ(h) we denote the domain space of a sesquilinear form h and in (1.1) we assume that Q(h) = Q(m). In the first part of the paper we show that (1.1) implies an estimate of the separation between “matching” eigensubspaces of H and A. To be more precise one of the typical situations is: Let 0 ≤ λ1(H) ≤ λ2(H) ≤ · · · ≤ λn(H) < D < λn+1(H) ≤ · · · (1.2) 0 ≤ λ1(M) ≤ λ2(M) ≤ · · · ≤ λn(M) < D < λn+1(M) ≤ · · · (1.3) be the eigenvalues of the operators H and M which satisfy (1.1) then ‖EH(D)− EM(D)‖ ≤ min {Dλn(H) D − λn(H) , √ Dλn(M) D − λn(M) } η. Such an estimate was implicit in [7]. We then generalize this inequality to hold both for the operator norm ‖ · ‖ and the Hilbert–Schmidt norm ||| · |||HS . We also allow that EH(D) and EM(D) be possibly infinite dimensional. For recent estimates of the separation between eigensubspaces see [10]. Date: February 8, 2007. 1991 Mathematics Subject Classification. 65F15, 49R50, 47A55, 35Pxx.