Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Summability in amenable semigroups


Author: Peter F. Mah
Journal: Trans. Amer. Math. Soc. 156 (1971), 391-403
MSC: Primary 40.50
DOI: https://doi.org/10.1090/S0002-9947-1971-0275013-X
MathSciNet review: 0275013
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A theory of summability is developed in amenable semigroups. We give necessary and (or) sufficient conditions for matrices to be almost regular, almost Schur, strongly regular, and almost strongly regular. In particular, when the amenable semigroup is the additive positive integers, our theorems yield those results of J. P. King, P. Schaefer and G. G. Lorentz for some of the matrices mentioned above.


References [Enhancements On Off] (What's this?)

  • [1] M. M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509-544. MR 19, 1067. MR 0092128 (19:1067c)
  • [2] -, Normed linear spaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Heft 21, Academic Press, New York; Springer-Verlag, Berlin, 1962. MR 26 #2847.
  • [3] E. E. Granirer, Extremely amenable semigroups, Math. Scand. 17 (1965), 177-197. MR 33 #5760. MR 0197595 (33:5760)
  • [4] -, Extremely amenable semigroups. II, Math. Scand. 20 (1967), 93-113. MR 35 #3422. MR 0212551 (35:3422)
  • [5] -, Functional analytic properties of extremely amenable semigroups, Trans. Amer. Math. Soc. 137 (1969), 53-75. MR 39 #765. MR 0239408 (39:765)
  • [6] -, On the invariant mean on topological semigroups and on topological groups, Pacific J. Math. 15 (1965), 107-140. MR 35 #286. MR 0209388 (35:286)
  • [7] P. R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, Chelsea, New York, 1951. MR 13, 563. MR 1653399 (99g:47001)
  • [8] J. P. King, Almost summable sequences, Proc. Amer. Math. Soc. 17 (1966), 1219-1225. MR 34 #1752. MR 0201872 (34:1752)
  • [9] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167-190. MR 10, 367. MR 0027868 (10:367e)
  • [10] T. Mitchell, Constant functions and left invariant means on semigroups, Trans. Amer. Math. Soc. 119 (1965), 224-261. MR 33 #1743. MR 0193523 (33:1743)
  • [11] -, Fixed points and multiplicative left invariant means, Trans. Amer. Math. Soc. 122 (1966), 195-202. MR 32 #7662. MR 0190249 (32:7662)
  • [12] P. Schaefer, Almost convergent and almost summable sequences, Proc. Amer. Math. Soc. 20 (1969), 51-54. MR 38 #3649. MR 0235340 (38:3649)
  • [13] J. Schur, Über lineare Transformation in der Theorie der unendlichen Reihen, J. Reine Angew. Math. 151 (1921), 79-111.
  • [14] O. Toeplitz, Über allgemeine lineare Mittelbildungen, Prace Mat.-Fiz. 22 (1911), 113-119.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 40.50

Retrieve articles in all journals with MSC: 40.50


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0275013-X
Keywords: Amenable semigroups, extremely amenable semigroups, almost convergent, almost convergent functions, invariant mean, almost regular matrices, almost Schur matrices, strongly regular matrices, almost strongly regular matrices
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society