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Functional analysis proofs of Abel’s theorems. (English) Zbl 1099.40003

The sequence of comlex numbers \(a_n\) is Abel convergent to \(a\) if \(\lim_{t\to 1^-}(1-t)\sum_{n=0}^{\infty}a_nt^n=a\). The series \(\sum_{n=0}^{\infty}a_n\) is (A) convergent to \(L\) if the sequence of its partial sums (A) converges to \(L\). The paper presents a proof of the well-known Abel theorem asserting that convergence implies (A) convergence. The proof is based on the theory of Hardy spaces in the unit disc.

MSC:

40A05 Convergence and divergence of series and sequences
47A15 Invariant subspaces of linear operators
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
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References:

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[2] R. E. Powell and S. M. Shah, Summability theory and applications, Van Nostrand Reinhold, London, 1972. · Zbl 0248.40001
[3] Eric Nordgren and Peter Rosenthal, Boundary values of Berezin symbols, Nonselfadjoint operators and related topics (Beer Sheva, 1992) Oper. Theory Adv. Appl., vol. 73, Birkhäuser, Basel, 1994, pp. 362 – 368. · Zbl 0874.47013
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