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A note on reductive operators


Author: Frank Gilfeather
Journal: Proc. Amer. Math. Soc. 44 (1974), 101-105
MSC: Primary 47A15
DOI: https://doi.org/10.1090/S0002-9939-1974-0341132-0
MathSciNet review: 0341132
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Abstract: A bounded linear operator $ A$ on a Hilbert space is called reductive if every invariant subspace of $ A$ reduces it. This paper gives examples of operators which give an affirmative answer to the reductive question: If $ A$ is reductive, then is $ A$ normal?


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  • [1] T. Ando, A note on invariant subspaces of a compact normal operator, Arch. Math. 14 (1963), 337-340. MR 27 #4073. MR 0154115 (27:4073)
  • [2] N. Aronszajn and K. T. Smith, Invariant subspaces of completely continuous operators, Ann. of Math, (2) 60 (1954), 345-350. MR 16, 488. MR 0065807 (16:488b)
  • [3] H. Behncke, Structure of certain nonnormal operators. II, Indiana Univ. Math. J. 22 (1972), 301-308. MR 0320788 (47:9322)
  • [4] A. R. Bernstein and A. Robinson, Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos, Pacific J. Math. 16 (1966), 421-431. MR 33 #1724. MR 0193504 (33:1724)
  • [5] N. Dunford and J. T. Schwartz, Linear operators. III, Wiley, Interscience, New York, 1971.
  • [6] J. A. Dyer, E. A. Pedersen and P. Porcelli, An equivalent formulation of the invariant subspace conjecture, Bull. Amer. Math. Soc. 78 (1972), 1020-1024. MR 0306947 (46:6068)
  • [7] F. Gilfeather, On the Suzuki structure theory for non self-adjoint operators on Hilbert space, Acta Sci. Math. (Szeged) 32 (1971), 239-249. MR 0305116 (46:4246)
  • [8] -, Operator valued roots of abelian analytic functions, Pacific J. Math. (to appear). MR 0367691 (51:3933)
  • [9] R. L. Moore, Properties of reductive operators, Notices Amer. Math. Soc. 20 (1973), A159.
  • [10] E. A. Nordgren, H. Radjavi and P. Rosenthal, Compact perturbations of normal operators with reducing invariant subspaces, Notices Amer. Math. Soc. 20 (1973), A155; and Address #509, American Mathematical Society 79th Annual Meeting, January 26, 1973.
  • [11] C. Pearcy and A. Shields, A survey of some recent results on invariant subspaces of operators on Banach spaces, Mimeographed notes, 1973.
  • [12] P. Rosenthal, Completely reducible operators, Proc. Amer. Math. Soc. 19 (1968), 826-830. MR 0231234 (37:6789)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0341132-0
Keywords: Reductive operator, invariant subspace
Article copyright: © Copyright 1974 American Mathematical Society

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