On the invertibility of isometric semigroup representations

C. Batty; D. Greenfield

Studia Mathematica (1994)

  • Volume: 110, Issue: 3, page 235-250
  • ISSN: 0039-3223

Abstract

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Let T be a representation of a suitable abelian semigroup S by isometries on a Banach space. We study the spectral conditions which will imply that T(s) is invertible for each s in S. On the way we analyse the relationship between the spectrum of T, Sp(T,S), and its unitary spectrum S p u ( T , S ) . For S = ℤ + n or ℝ + n , we establish connections with polynomial convexity.

How to cite

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Batty, C., and Greenfield, D.. "On the invertibility of isometric semigroup representations." Studia Mathematica 110.3 (1994): 235-250. <http://eudml.org/doc/216111>.

@article{Batty1994,
abstract = {Let T be a representation of a suitable abelian semigroup S by isometries on a Banach space. We study the spectral conditions which will imply that T(s) is invertible for each s in S. On the way we analyse the relationship between the spectrum of T, Sp(T,S), and its unitary spectrum $Sp_\{u\}(T,S)$. For $S = ℤ^\{n\}_\{+\}$ or $ℝ^\{n\}_\{+\}$, we establish connections with polynomial convexity.},
author = {Batty, C., Greenfield, D.},
journal = {Studia Mathematica},
keywords = {semigroup; isometric representation; spectrum; polynomial convexity; representation of a suitable Abelian semigroup; isometries on a Banach space; spectral conditions; unitary spectrum},
language = {eng},
number = {3},
pages = {235-250},
title = {On the invertibility of isometric semigroup representations},
url = {http://eudml.org/doc/216111},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Batty, C.
AU - Greenfield, D.
TI - On the invertibility of isometric semigroup representations
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 3
SP - 235
EP - 250
AB - Let T be a representation of a suitable abelian semigroup S by isometries on a Banach space. We study the spectral conditions which will imply that T(s) is invertible for each s in S. On the way we analyse the relationship between the spectrum of T, Sp(T,S), and its unitary spectrum $Sp_{u}(T,S)$. For $S = ℤ^{n}_{+}$ or $ℝ^{n}_{+}$, we establish connections with polynomial convexity.
LA - eng
KW - semigroup; isometric representation; spectrum; polynomial convexity; representation of a suitable Abelian semigroup; isometries on a Banach space; spectral conditions; unitary spectrum
UR - http://eudml.org/doc/216111
ER -

References

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  2. [2] H. Alexander, Totally real sets in C 2 , Proc. Amer. Math. Soc. 111 (1991), 131-133. Zbl0734.32008
  3. [3] W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc. 306 (1988), 837-852. Zbl0652.47022
  4. [4] C. J. K. Batty and Vũ Quôc Phóng, Stability of strongly continuous representations of abelian semigroups, Math. Z. (1992), 75-88. 
  5. [5] R. G. Douglas, On extending commutative semigroups of operators, Bull. London Math. Soc. 1 (1969), 157-159. Zbl0187.06501
  6. [6] Yu. I. Lyubich, Introduction to the Theory of Banach Representations of Groups, Birkhäuser, Basel, 1988. 
  7. [7] Yu. I. Lyubich and Vũ Quôc Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Math. 88 (1988), 37-42. Zbl0639.34050
  8. [8] G. K. Pedersen, C*-algebras and their Automorphism Groups, Academic Press, London, 1979. Zbl0416.46043
  9. [9] Vũ Quôc Phóng and Yu. I. Lyubich, A spectral criterion for asymptotic almost periodicity for uniformly continuous representations of abelian semigroups, Teor. Funktsiǐ Funktsional. Anal. i Prilozhen. 50 (1988), 38-43 (in Russian); English transl.: J. Soviet Math. 49 (1990), 1263-1266. 
  10. [10] W. Rudin, Fourier Analysis on Groups, Wiley, New York, 1962. Zbl0107.09603
  11. [11] G. M. Sklyar and V. A. Shirman, On the asymptotic stability of a linear differential equation in a Banach space, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 37 (1982), 127-132 (in Russian). Zbl0521.34063
  12. [12] G. Stolzenberg, Polynomially and rationally convex sets, Acta Math. 109 (1963), 259-289. Zbl0122.08404
  13. [13] J. Wermer, Banach Algebras and Several Complex Variables, Springer, New York, 1976. Zbl0336.46055

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