Spectral properties of linear operators for which and commute

Authors:
Stephen L. Campbell and Ralph Gellar

Journal:
Proc. Amer. Math. Soc. **60** (1976), 197-202

MSC:
Primary 47B20

MathSciNet review:
0417841

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Abstract: The class of linear operators for which and commute is studied. It is shown that such operators are normaloid. If *T* is also completely nonnormal, then . Also, isolated points of are reducing eigenvalues. Finally, if is a subset of either a vertical line or the real axis, then *T* is normal.

**[1]**S. K. Berberian,*Some conditions on an operator implying normality*, Math. Ann.**184**(1969/1970), 188–192. MR**0256205****[2]**S. K. Berberian,*Some conditions on an operator implying normality. II*, Proc. Amer. Math. Soc.**26**(1970), 277–281. MR**0265975**, 10.1090/S0002-9939-1970-0265975-3**[3]**Arlen Brown,*On a class of operators*, Proc. Amer. Math. Soc.**4**(1953), 723–728. MR**0059483**, 10.1090/S0002-9939-1953-0059483-2**[4]**Stephen L. Campbell,*Linear operators for which 𝑇*𝑇 and 𝑇+𝑇* commute*, Pacific J. Math.**61**(1975), no. 1, 53–57. MR**0405168****[5]**Stephen L. Campbell,*Operator-valued inner functions analytic on the closed disc. II*, Pacific J. Math.**60**(1976), no. 2, 37–49. MR**0428078****[6]**Mary R. Embry,*A connection between commutativity and separation of spectra of operators*, Acta Sci. Math. (Szeged)**32**(1971), 235–237. MR**0303321****[7]**Paul R. Halmos,*A Hilbert space problem book*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0208368****[8]**Bernard B. Morrel,*A decomposition for some operators*, Indiana Univ. Math. J.**23**(1973/74), 497–511. MR**0343079**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0417841-3

Keywords:
Operator such that and commute,
spectrum,
normaloid operator,
spectraloid operator,
isoloid operator

Article copyright:
© Copyright 1976
American Mathematical Society