On the mean ergodic theorem for Cesàro bounded operators

On the Mean Ergodic Theorem for Subsequences

With these assumptions we have T defined for every integer n as a 1-1, onto, bimeasurable transformation. Henceforth we shall assume that every set considered is measurable, i.e. an element of a. We shall say that P is invariant if P(A) =P(TA) for every set A, P is ergodic if P is invariant and if P(U^L_oo TA) = 1 for every set A for which P(A) > 0 , and finally P is strongly mixing if P is inv...

A Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition

Spectral Theorem for Bounded Self-adjoint Operators

Diagonalization is one of the most important topics one learns in an elementary linear algebra course. Unfortunately, it only works on finite dimensional vector spaces, where linear operators can be represented by finite matrices. Later, one encounters infinite dimensional vector spaces (spaces of sequences, for example), where linear operators can be thought of as ”infinite matrices”. Extendin...

A Morse Index Theorem for Elliptic Operators on Bounded Domains

We consider a second-order, selfadjoint elliptic operator L on a smooth one-parameter family of domains {Ωt}t∈[a,b] with no assumptions on the geometry of the Ωt’s. It is shown that the Morse index of L can be equated with the Maslov index of an appropriately defined path in a symplectic Hilbert space constructed on the boundary of Ωb. Our result is valid for a wide variety of boundary conditio...

Mean Ergodic Theorems for C0 Semigroups of Continuous Linear Operators

In this paper we obtained mean ergodic theorems for semigroups of bounded linear or continuous affine linear operators on a Banach space under non-power bounded conditions. We then apply them to the wave equation and the system of elasticity to show that the mean of their solutions converges to their equilibriums.