On asymptotic properties of positive operators on banach lattices

---

Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2006

Öğrenci: ALİ BİNHADJAH

Danışman: EDUARD EMELYANOV

Özet:

In this thesis, we study two problems. The first one is the renorming problem in Banach lattices. We state the problem and give some known results related to it. Then we pass to construct a positive doubly power bounded operator with a nonpositive inverse on an infinite dimensional AL-space which generalizes the result of [10]. The second problem is related to the mean ergodicity of positive operators on KBspaces. We prove that any positive power bounded operator T in a KB-space E which satisfies lim n!1 dist 1 n n1 Xk=0 Tkx, [g, g] + BE = 0 (8x 2 E, kxk 1), ( ) where BE is the unit ball of E, g 2 E+, and 0 < 1, is mean ergodic and its fixed space Fix(T) is finite dimensional. This generalizes the main result of [12]. Moreover, under the assumption that E is a -Dedekind complete Banach lattice, we prove that if, for any positive power bounded operator T, the condition ( ) implies that T is mean ergodic then E is a KB-space.