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On Statistical and Ideal Convergence of Sequences of Bounded Linear Operators

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Abstract (2. Language): 
Let (An) be a sequence of bounded linear operators from a separable Banach space X into a Banach space Y . Suppose that  is a countable fundamental set of X and the ideal I of subsets of N has property (AP). The sequence (An) is said to be b∗I -convergent if it is pointwise I -convergent and there exists an index set K such that N \ K ∈ I and (Ak x)k∈K is bounded for any x ∈ X. We prove that the sequence (An) is b∗I -convergent if and only if (kAnk) is I -bounded and (Anφ) is I -convergent for any φ ∈ . Applications of this Banach–Steinhaus type theorem are related to some sequence-to-sequence matrix transformations and to the weak I -convergence in Banach spaces.
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