On an analog of the M.G. Krein theorem for measurable operators - A.M. Bikchentaev (Учёные записки Казанского университета: Серия Физико-математические науки)
Let M be a von Neumann algebra of operators on a Hilbert space H and τ be a faithful normal semifinite trace on M. Let μt(T), t > 0, be a rearrangement of a τ-measurable operator T. Let us consider a τ-measurable operator A, such that μt(A) > 0 for all t > 0 and assume that μ2t(A) / μt(A) →1 as t→∞. Let a τ-compact operator S be so that the operator I+S is right invertible, where I is the unit of M. Then, for a τ-measurable operator B, such that A=B(I+S), we have μt(A) / μt(B) →1 as t→∞. It is an analog of the M.G. Krein theorem (for M=B(H) and τ=tr, theorem 11.4, ch. V [Gohberg I.C., Krein M.G. Introduction to the theory of linear nonselfadjoint operators. In: Translations of Mathematical Monographs. Vol. 18. Providence, R.I., Amer. Math. Soc., 1969. 378 p.] for τ-measurable operators.