Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-27T20:34:23.918Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal Extensions of Representations of Abelian Semigroups

Published online by Cambridge University Press:  20 November 2018

Boyu Li
Boyu Li*
Affiliation:
Pure Mathematics Department, University of Waterloo, Waterloo, ON, N2L–3G1 e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A commuting family of subnormal operators need not have a commuting normal extension. We study when a representation of an abelian semigroup can be extended to a normal representation, and show that it suffices to extend the set of generators to commuting normals. We also extend a result due to Athavale to representations on abelian lattice ordered semigroups.

Keywords

47B2047A2047D03subnormal operatornormal extensionregular dilationlattice ordered semigroup
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 3 , 01 September 2016 , pp. 564 - 574
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Abrahamse, M. B., Commuting subnormal operators.Illinois J. Math. 22(1978), no. 1,171-176.Google Scholar
[2] Agler, J., Hypercontractions and subnormality. J. Operator Theory 13(1985), 203–217.Google Scholar
[3] Anderson, M. E. and Feil, T. H., Lattice-ordered groups: an introduction, volume 4. Springer Science & Business Media, 2012.Google Scholar
[4] Athavale, A., Holomorphic kernels and commuting operators. Trans. Amer. Math. Soc. 304(1987), no. 1, 101–110. http://dx.doi.org/10.2307/2000706 Google Scholar
[5] Athavale, A., Relating the normal extension and the regular unitary dilation of a subnormal tuple of contractions. Acta Sci. Math.(Szeged) 56(1992), no. 1-2, 121–124.Google Scholar
[6] Athavale, A. and Pedersen, S., Moment problems and subnormality. J. Math. Anal. Appl. 146(1990), no. 2, 434–441. http://dx.doi.org/10.101 6/0022-247X(90)90314-6 Google Scholar
[7] Brehmer, S., Ùber vetauschbare Kontraktionen des Hilbertschen Raumes. Acta Sci. Math. Szeged 22(1961), 106–111.Google Scholar
[8] Broschinski, A., Eigenvalues ofToeplitz operators on the annulus and Neil algebra. Complex Anal. Oper. Theory 8(2014), no. 5,1037-1059. http://dx.doi.org/10.1007/s11785-013-0331-5 Google Scholar
[9] Conway, J. B., The theory of subnormal operators. Mathematical Surveys and Monographs, 36, American Mathematical Society, Providence, 1991. http://dx.doi.org/10.1090/surv/036 Google Scholar
[10] Dritschel, M. A., Jury, M. T., and McCullough, S., Dilations and constrained algebras.2013. arxiv:1305.4272Google Scholar
[11] Itô, T., On the commutative family of subnormal operators. J. Fac. Sci. Hokkaido Univ. Ser. I 14(1958), 1–15.Google Scholar
[12] Li, B., Regular representations of lattice ordered semigroups. 2015. arxiv:1503.03046Google Scholar
[13] Lubin, A., A subnormal semigroup without normal extension. Proc. Amer. Math. Soc. 68(1978), no. 2, 176–178.Google Scholar
[14] Sz.-Nagy, B., Extensions of linear transformations in Hilbert space which extend beyond this space. In: Functional analysis, Frederic Ungar Pub. Co., 1960.Google Scholar