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Pointwise limit theorem for a class of unbounded operators in r -spaces

Ryszard Jajte — 2007

Studia Mathematica

We distinguish a class of unbounded operators in r , r ≥ 1, related to the self-adjoint operators in ². For these operators we prove a kind of individual ergodic theorem, replacing the classical Cesàro averages by Borel summability. The result is equivalent to a version of Gaposhkin’s criterion for the a.e. convergence of operators. In the proof, the theory of martingales and interpolation in r -spaces are applied.

Convergence of orthogonal series of projections in Banach spaces

Ryszard JajteAdam Paszkiewicz — 1997

Annales Polonici Mathematici

For a sequence ( A j ) of mutually orthogonal projections in a Banach space, we discuss all possible limits of the sums S n = j = 1 n A j in a “strong” sense. Those limits turn out to be some special idempotent operators (unbounded, in general). In the case of X = L₂(Ω,μ), an arbitrary unbounded closed and densely defined operator A in X may be the μ-almost sure limit of S n (i.e. S n f A f μ-a.e. for all f ∈ (A)).

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