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Spectral decompositions, ergodic averages, and the Hilbert transform

Earl Berkson, T. A. Gillespie (2001)

Studia Mathematica

Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ₙ ( U ) n = 1 ∞ the sequence of (C,2) weighted discrete ergodic averages of U, that is, ₙ ( U ) = 1 / n ∑ 0 < | k | ≤ n ( 1 - | k | / ( n + 1 ) ) U k . We show that this sequence ₙ ( U ) n = 1 ∞ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral theory and...

Summable families in nuclear groups

Wojciech Banaszczyk (1993)

Studia Mathematica

Nuclear groups form a class of abelian topological groups which contains LCA groups and nuclear locally convex spaces, and is closed with respect to certain natural operations. In nuclear locally convex spaces, weakly summable families are strongly summable, and strongly summable are absolutely summable. It is shown that these theorems can be generalized in a natural way to nuclear groups.

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