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.


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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