## 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)

### Keywords:

Abel convergent; Berezin symbol; diagonal operator
Full Text:

### References:

 [1] A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. · Zbl 0085.05601 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.