We consider a continuous operator T : E → X where E is a Banach lattice and X is a Banach space. We characterize the b-weak compactness of T in terms of its mapping properties. (2000). 47B60, 47B10, 47B07.
Mathematics Subject ClassificationLet E be a Riesz space. E ∼ and E ∼∼ will denote the order dual and order bidual of E respectively. The canonical embedding Q E : E → E ∼∼ is defined byx is an order bounded and order continuous linear functional on E ∼ . The canonical embedding Q E is a lattice homomorphism. If E ∼ separates the points of E, Q E is also one-to-one and hence, E can be considered as a Riesz subspace of E ∼∼ . Since all Banach lattices have separating order duals, we will not distinguish a Banach lattice E and its image in E , the bidual of E. We will assume all Riesz spaces considered in this note have separating order duals.
Definition. Let A be a subset of the Riesz space E. If Q E (A) is order bounded inClearly, every order bounded subset of E is a b-order bounded subset of E. There are b-order bounded subsets of E which are far away from being order bounded. The canonical basis vectors {e n } of c 0 is b-order bounded but clearly not an order bounded subset of c 0 .The notion of b-order bounded subsets of a Riesz space is instrumental in distinguishing a family of Riesz spaces.
Definition. A Riesz space E is said to have the b-property if every b-order bounded subset of E is order bounded in E.b-order boundedness was introduced in [3]. b-order boundedness and Riesz spaces with the b-property were also investigated in [4,5,6].
A Riesz space E is said to have b-property if each subset which is order bounded in E ∼∼ is order bounded in E. The relationship between b-property and completeness, being a retract and the absolute weak topology |σ| (E ∼ , E) is studied. Perfect Riesz spaces are characterized in terms of b-property. It is shown that b-property coincides with the Levi property in Dedekind complete Frechet lattices.
Mathematics Subject Classification (2000). Primary 46A40.
We prove that any positive power bounded operator T in a KB-space E which satisfieswhere B E is the unit ball of E, g ∈ E + , and 0 η < 1, is mean ergodic and its fixed space Fix (T ) is finite dimensional. This generalizes the main result of [E.Yu. Emelyanov, M.P.H. Wolff, Mean lower bounds for Markov operators, Ann. Polon. Math. 83 (2004) 11-19].Moreover, under the assumption that E is a σ -Dedekind complete Banach lattice, we prove that if, for any positive power bounded operator T , the condition (1) implies that T is mean ergodic then E is a KB-space.
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