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Báo cáo toán học: "Ergodic theory and the functional equation (I - T)x = y "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Ergodic lý thuyết và phương trình chức năng (I - T) x = y. | J. OPERATOR THEORY 10 1983 153-166 Copyright by INCREST 1983 ERGODIC THEORY AND THE FUNCTIONAL EQUATION 7 - T x y MICHAEL LIN and ROBERT SINE The problem of solving the functional equation Z T x y for a given linear operator T on a Banach space X and a given y e X appears in many areas of analysis and probability. The well-known Neumann series gives ự T -1 when T 1. When II 7 11 J the problem is first to know if y e I T X and then to find the solution X. The solution is usually found using an iterative procedure see 4 5 6 16 . We are interested in the convergence of n k - 1 x - n ỵ y y TJy to the solution X and obtain the precise necessary and sufficient k .1 -0 conditions Corollary 3 . The necessa y condition sup y TJy oo is shown to be sufficient if T for some m 0 is weakly compact. An example shows that otherwise the condition need not be sufficient. The reflexive case appears in 1 2 3 . We then solve the problem of existence in the case of a dual operator on a dual space obtaining as a corollary an application to Markov operators. Next we look at the same problem for Tf s f Os where 7 is induced on a suitable function space by a measurable map 0. A new ergodic proof for 0 a minimal continuous map of a Hausdorff space is given. Finally we obtain results for positive conservative contractions Markov operators on Lỵ S z Ì . In that case we look also at solutions which are finite a.e. though not necessary in Z.J. For the general Banach space approach we need the mean ergodic theorem-. If Tfn 1 n 1 0 strongly and sup y TJ n jto oo then 1 n-l Ì _ . - X - - y TJX converges y Ty y Z T X. n J-X J 154 MICHAEL LIN and ROBERT SINE We call T mean ergodic if the above subspace is all of X. We mention the uniform ergodic theorem 19 M 1 T X is closed converses uniformly. k- 0 Ị n k--ỉ In that case I T is invertible on Z T X and - yj J T converges n k 1 j 0 uniformly to I T -1 on Z T X which is a generalization of the Neumann series theorem. Theorem 1. Let T be mean .