Steinhaus type theorems for summability matrices

Maddox, I. J.

doi:10.1090/S0002-9939-1974-0364938-0

Steinhaus type theorems for summability matrices
HTML articles powered by AMS MathViewer

by I. J. Maddox PDF

Proc. Amer. Math. Soc. 45 (1974), 209-213 Request permission

Abstract:

Necessary and sufficient conditions are given for an infinite matrix to sum all bounded strongly summable sequences. It is shown that the Borel matrix does not sum all such sequences. A corollary is that the bounded summability field of the Borel method is strictly contained in that of the $(C,1)$ method. Also, it is proved that no coregular matrix can almost sum all bounded sequences—a generalization of Steinhaus’ theorem.

References

D. Borwein, Linear functionals connected with strong Cesàro summability, J. London Math. Soc. 40 (1965), 628–634. MR 185426, DOI 10.1112/jlms/s1-40.1.628
G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. MR 0030620
J. P. King, Almost summable sequences, Proc. Amer. Math. Soc. 17 (1966), 1219–1225. MR 201872, DOI 10.1090/S0002-9939-1966-0201872-6
B. Kuttner, Note on strong summability, J. London Math. Soc. 21 (1946), 118–122. MR 19141, DOI 10.1007/BF01897108
G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167–190. MR 27868, DOI 10.1007/BF02393648
I. J. Maddox, On Kuttner’s theorem, J. London Math. Soc. 43 (1968), 285–290. MR 225044, DOI 10.1112/jlms/s1-43.1.285
I. J. Maddox, Elements of functional analysis, Cambridge University Press, London-New York, 1970. MR 0390692