Individual Ergodic Theorem for Unitary Maps of Random Matrices
نویسندگان
چکیده
Using simple techniques of finite von Neumann algebras, we prove a limit theorem for random matrices. 1. Notation and main result 1.1. Let (Ω,F , μ) be a probability space, and let H denote the space of all d × d matrices with entries in the complex space L2(Ω, μ). H is a Hilbert space with the inner product (1) 〈AB〉 = Φ(AB∗), where φ(A) = ∫ Ω tr A(ω)μ(dμ), and tr denotes the normalized trace on the algebra Md of d×d complex matrices. A matrix A with F -measurable entries (random matrix) is said to be positive (A ≥ 0) if for μ-almost all ω ∈ Ω, A(ω) is positive definite in C, i.e., (A(ω)λ, λ) ≥ 0 for λ ∈ C. A ≥ B means A − B ≥ 0. A map α : H → H is said to be positive if αA ≥ 0 for A ≥ 0. Throughout the paper 1 denotes the unit random matrix, i.e., the diagonal matrix with the entries aik = δ i μ-a.e. The main goal of the paper is the following individual ergodic theorem. 1.2. Theorem. Let α be a positive unitary operator in H, satisfying the condition (2) 1 n n ∑ r=−n α1 ≤ Y, n = 1, 2, . . . , for some random matrix Y with φ(Y ) <∞. Let us put (3) Sn = 1 n n−1 ∑ k=0 α. Then, for each A ∈ H, (4) ||(SnA)(ω)− Ã(ω)|| → 0 μ-a.e., || · || being an arbitrary fixed norm in Md; here à is given by the mean ergodic theorem for α. Received by the editors August 12, 2002 and, in revised form, February 26, 2003. 2000 Mathematics Subject Classification. Primary 60F15, 46L10.
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