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Semi-Groups of Maps in a Locally Compact Space

Published online by Cambridge University Press:  20 November 2018

J. R. Dorroh
J. R. Dorroh*
Affiliation:
Louisiana State University, Baton Rouge
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Suppose that S is a locally compact Hausdorff space. A one-parameter semi-group of maps in S is a family {ϕt; t ⩾ 0} of continuous functions from S into S satisfying

  1. (i) ϕt0ϕu = ϕt+u for t, u ⩾ 0, where the circle denotes composition, and

  2. (ii) ϕ0 = e, the identity map on S.

    A semi-group {ϕt} of maps in S is said to be

  3. (iii) of class (C0) if ϕt(x) → x as t → 0 for each x in S,

  4. (iv) separately continuous if the function t → ϕt(x) is continuous on [0, ∞) for each x in S, and

  5. (v) doubly continuous if the function (t, x) → (ϕt(x) is continuous on [0, ∞) x S.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 688 - 696
Copyright
Copyright © Canadian Mathematical Society 1967

References

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