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Joint spectra of commuting normal operators on Banach spaces. (English) Zbl 0689.47001
Let \(T=(T_ 1,...,T_ n)\) be an n-tuple of operators on a complex Banach space X. T is called strongly commuting if there are operators \(U_ j\), \(V_ j\) with real spectra such that \(T_ j=U_ j+iV_ j\) and \((U_ 1,...,U_ n,V_ 1,...,V_ n)\) is a commuting 2n-tuple. The author proves that the Taylor joint spectrum \(\sigma\) (T) of a strong commuting n-tuple coincides with the approximate point spectrum \(\sigma_{\pi}(T)\). For a commuting n-tuple T of normal operators, which is strongly commuting by the Fuglede theorem, the author proves moreover that co(\(\sigma\) (T)) is equal to the closure of the spatial joint numerical range \(\overline{V(T)}\). Another result is that an extreme point \(z\in \overline{V(T)}\) with \(z\in V(T)\) is a joint eigenvalue of T.
Reviewer: M.Lesch

MSC:
47A10 Spectrum, resolvent
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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References:
[1] Vasilescu, Rev. Roum. Math. Pures Appl. 22 pp 1003– (1977)
[2] Wrobel, Glasgow Math. J. 30 pp 145– (1988)
[3] DOI: 10.1007/BF02392329 · Zbl 0233.47025 · doi:10.1007/BF02392329
[4] DOI: 10.2307/1998391 · Zbl 0457.47017 · doi:10.2307/1998391
[5] DOI: 10.1512/iumj.1980.29.29028 · Zbl 0414.47030 · doi:10.1512/iumj.1980.29.29028
[6] DOI: 10.1090/S0002-9904-1974-13481-6 · Zbl 0276.47001 · doi:10.1090/S0002-9904-1974-13481-6
[7] Chō, Glasgow Math. J. 28 pp 69– (1986)
[8] Chō, Studia Math. 80 pp 245– (1984)
[9] Ceausescu, Studia Math. 62 pp 305– (1978)
[10] DOI: 10.2307/2042460 · Zbl 0403.47001 · doi:10.2307/2042460
[11] Bonsall, Numerical ranges II (1973) · doi:10.1017/CBO9780511662515
[12] Bonsall, Numerical ranges of operators on normed spaces and elements of normed algebras (1971) · Zbl 0207.44802 · doi:10.1017/CBO9781107359895
[13] DOI: 10.1016/0022-1236(70)90055-8 · Zbl 0233.47024 · doi:10.1016/0022-1236(70)90055-8
[14] Slodkowski, Studia Math. 50 pp 127– (1974)
[15] DOI: 10.1017/S0305004100062411 · Zbl 0558.47020 · doi:10.1017/S0305004100062411
[16] Mattila, Ann. Acad. Sci. Fnn. A I, Math. Dissertations 19 (1978)
[17] Mattila, Math. Scand. 43 pp 363– (1978) · Zbl 0387.47001 · doi:10.7146/math.scand.a-11790
[18] DOI: 10.1090/S0002-9904-1972-13066-0 · Zbl 0255.47053 · doi:10.1090/S0002-9904-1972-13066-0
[19] DOI: 10.2307/2044243 · Zbl 0476.47017 · doi:10.2307/2044243
[20] Vasilescu, Studia Math. 62 pp 203– (1978)
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