Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-23T06:56:27.707Z Has data issue: false hasContentIssue false
  • English
  • Français

On joint spectra of non-commuting normal operators

Published online by Cambridge University Press:  17 April 2009

Alan J. Pryde  and
Andrzej Sołtysiak
Alan J. Pryde
Affiliation:
Department of Mathematics, Monash University, Clayton Vic 3168, Australia
Andrzej Sołtysiak
Affiliation:
Institute of Mathematics A. Mickiewicz University Matejki, 48/49, 60769 Poznań, Poland
Rights & Permissions [Opens in a new window]

Extract

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.

The purpose of the paper is to show that the Harte spectrum and the bicommutant spectrum of an arbitrary n−tuple of normal Hilbert space operators can be obtained from the spectral set γ introduced by McIntosh and Pryde. It is also proved that many commonly used joint spectra of an n−tuple of normal m by m matrices are equal. These results are non-commutative variants of some theorems proved by McIntosh, Pryde, and Ricker for commuting sets of operators.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 1 , August 1993 , pp. 163 - 170
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Curto, R.E., ‘Connections between Harte and Taylor spectra’, Rev. Roumaine Math. Pures Appl. 31 (1968), 203–215.Google Scholar
[2]Dash, A.T., ‘Joint essential spectra’, Pacific J. Math 64 (1976), 119–128.CrossRefGoogle Scholar
[3]Fong, C.-K. and Sołtysiak, A., ‘Existence of a multiplicative functional and joint spectra’, Studia Math. 81 (1985), 213–220.CrossRefGoogle Scholar
[4]Greiner, G. and Ricker, W.J., ‘Joint spectral sets and commutativity of systems of 2 × 2 selfadjoint matrices’. Preprint.Google Scholar
[5]Halmos, P.R., Finite-dimensional vector spaces (D. Van Nostrand, Princeton, 1958).Google Scholar
[6]Harte, R.E., ‘Spectral mapping theorems’, Proc. Roy. Irish Acad. Sect. A 72 (1972), 89–107.Google Scholar
[7]McIntosh, A.G.R. and Pryde, A.J., ‘The solutions of systems of operator equations using Clifford algebras’, Proc. Centre Math. Anal. Austral. Nat. Univ. 9 (1985), 212–220.Google Scholar
[8]McIntosh, A.G.R. and Pryde, A.J., ‘A functional calculus for several commuting operators’, Indiana Univ. Math. J. 36 (1987), 421–439.CrossRefGoogle Scholar
[9]McIntosh, A.G.R., Pryde, A.J. and Ricker, W.J., ‘Comparison of joint spectra for certain classes of commuting operators’, Studia Math. 88 (1988), 23–36.CrossRefGoogle Scholar
[10]Müller, V. and Sołtysiak, A., ‘On the largest generalized joint spectrum’, Comment. Math. Univ. Carolin. 29 (1988), 255–259.Google Scholar
[11]Müller, V. and Sołtysiak, A., ‘Spectrum of generators of a noncommutative Banach algebra’, Studia Math 93 (1989), 87–95.CrossRefGoogle Scholar
[12]Taylor, J.L., ‘A joint spectrum for several commuting operators’, J. Fund. Anal. 6 (1970), 172–191.CrossRefGoogle Scholar