Tensor products and almost periodicity
Author:
Hugo D. Junghenn
Journal:
Proc. Amer. Math. Soc. 43 (1974), 99-105
MSC:
Primary 47D05; Secondary 46M05
DOI:
https://doi.org/10.1090/S0002-9939-1974-0365223-3
MathSciNet review:
0365223
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Abstract | References | Similar Articles | Additional Information
Abstract: Let and
be locally convex spaces and
their completed
-tensor product. It is shown that if
and
are weakly almost periodic equicontinuous semigroups of operators on
and
respectively, then, under mild restrictions on
or
is a weakly almost periodic equicontinuous semigroup of operators on
, and the almost periodic and flight vector subspaces of
are related in a natural way to the corresponding subspaces of
and
via the
-tensor product. Furthermore, if
and
both decompose into a direct sum of these subspaces then so does
.
- [1] J. F. Berglund and K. H. Hofmann, Compact semitopological semigroups and weakly almost periodic functions, Lecture Notes in Mathematics, No. 42, Springer-Verlag, Berlin-New York, 1967. MR 0223483
- [2] K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97. MR 131784, https://doi.org/10.1007/BF02559535
- [3] W. F. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math. Soc. 67 (1949), 217–240. MR 36455, https://doi.org/10.1090/S0002-9947-1949-0036455-9
- [4] Robert Ellis, Locally compact transformation groups, Duke Math. J. 24 (1957), 119–125. MR 88674
- [5] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), Chapter 1: 196 pp.; Chapter 2: 140 (French). MR 0075539
- [6] Konrad Jacobs, Ergodentheorie und fastperiodische Funktionen auf Halbgruppen, Math. Z. 64 (1956), 298–338 (German). MR 77092, https://doi.org/10.1007/BF01166575
- [7] H. Junghenn, Almost periodic compactifications and applications to one-parameter semigroups, Thesis, The George Washington University, Washington, D.C., 1971.
- [8] Helmut H. Schaefer, Topological vector spaces, Springer-Verlag, New York-Berlin, 1971. Third printing corrected; Graduate Texts in Mathematics, Vol. 3. MR 0342978
- [9] Robert Schatten, A Theory of Cross-Spaces, Annals of Mathematics Studies, no. 26, Princeton University Press, Princeton, N. J., 1950. MR 0036935
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1974-0365223-3
Keywords:
Semigroup of operators,
-tensor product,
weakly almost periodic,
almost periodic,
flight vector,
reversible vector
Article copyright:
© Copyright 1974
American Mathematical Society