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Linear operators for which $T^{\ast } T$ and $TT^{\ast }$ commute


Author: Stephen L. Campbell
Journal: Proc. Amer. Math. Soc. 34 (1972), 177-180
MSC: Primary 47B99
DOI: https://doi.org/10.1090/S0002-9939-1972-0295124-9
MathSciNet review: 0295124
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Abstract: Linear operators T for which ${T^\ast }T$ and $T{T^\ast }$ commute are studied. Examples are given to show that this class of operators is distinct from several other operator classes. It is proven that if ${T^\ast }T$ and $T{T^\ast }$ commute and T is hyponormal, then T has an invariant subspace. A generalization of this theorem is given.


References [Enhancements On Off] (What's this?)

  • Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
  • Peter A. Fillmore, Notes on operator theory, Van Nostrand Reinhold Mathematical Studies, No. 30, Van Nostrand Reinhold Co., New York-London-Melbourne, 1970. MR 0257765
  • Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368

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Additional Information

Keywords: Hyponormal operators, invariant subspaces, operators for which <IMG WIDTH="44" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${T^\ast }T$"> and <IMG WIDTH="44" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$T{T^\ast }$"> commute
Article copyright: © Copyright 1972 American Mathematical Society