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We investigate countably convex $G_{\delta }$ subsets of Banach spaces. A subset of a linear space is countably convex if it can be represented as a countable union of convex sets. A known sufficient condition for countable convexity of an arbitrary subset of a separable normed space is that it does not contain a semi-clique [9]. A semi-clique in a set S is a subset P ⊆ S so that for every x ∈ P and open neighborhood u of x there exists a finite set X ⊆ P ∩ u such that conv(X) ⊈ S. For closed sets this condition...

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I-T)X=z\in X:su{p}_n\parallel {\sum }_{k=0}^n{T}^{k}z\parallel <\infty$. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{(I-T)X}$ contains an isomorphic copy of an infinite-dimensional dual Banach space. Consequently, if X is separable and does not contain isomorphic copies of infinite-dimensional dual Banach spaces, then (*) is equivalent to uniform ergodicity. As an application, sufficient conditions for uniform ergodicity of irreducible Markov chains...

We show that a Banach space X has an infinite dimensional reflexive subspace (quotient) if and only if there exist a Banach space Z and a non-isomorphic one-to-one (dense range) Tauberian (co-Tauberian) operator form X to Z (Z to X). We also give necessary and sufficient condition for the existence of a Tauberian operator from a separable Banach space to c which in turn generalizes a result of Johnson and Rosenthal. Another application of our result shows that if X** is separable, then there exists...

Let X be a Banach space with a basis. We prove that X is reflexive if and only if every power-bounded linear operator T satisfies Browder’s equality $x\in X:su{p}_n\left|\right|{\sum }_{k=1}^n{T}^{k}x\left|\right|<\infty$ = (I-T)X$.$We then deduce that X (with a basis) is reflexive if and only if every strongly continuous bounded semigroup $T_t:t\ge 0$ with generator A satisfies $AX=x\in X:su{p}_{s>0}\left|\right|{\int }_0^s{T}_{t}xdt\left|\right|<\infty$. The range (I-T)X (respectively, AX for continuous time) is the space of x ∈ X for which Poisson’s equation (I-T)y = x (Ay = x in continuous time) has a solution y ∈ X; the above equalities for the ranges...

Sea T un operador lineal acotado e inyectivo de un espacio de Banach X en un espacio de Hilbert H con rango denso y sea {x} ⊂ X una sucesión tal que {Tx} es ortogonal. Se estudian propiedades de {Tx} dependientes de propiedades de {x}. También se estudia la ""situación opuesta"", es decir, la acción de un operador T : H → X sobre sucesiones ortogonales.