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The canonical form of a scalar operator on a Banach space


Authors: G. D. Faulkner and J. E. Huneycutt
Journal: Proc. Amer. Math. Soc. 71 (1978), 81-84
MSC: Primary 47B40; Secondary 46G10
MathSciNet review: 0487577
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Abstract: Let $ A = \smallint \lambda dE(\lambda )$ be a scalar operator on a Banach space X. If there exists a vector $ g \in X$ such that the closed convex hull of the range of the vector measure $ \mu ( \cdot ) = E( \cdot )g$ has nonvoid interior, then A is similar to the operator $ Qf(\lambda ) = \lambda f(\lambda )$ on a quotient space of a suitably constructed $ {\mathcal{L}^\infty }$ space.


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

  • [1] N. I. Akhiezer and I. M. Glazman, The theory of linear operators in Hilbert space, Ungar, New York, 1963.
  • [2] N. Dunford and J. T. Schwartz, Linear operators, Interscience, New York, 1963. MR 0188745 (32:6181)
  • [3] I. Kluvanek and G. Knowles, Vector measures and control systems, New York, 1976. MR 0499068 (58:17033)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0487577-3
Keywords: Scalar operator, vector measures
Article copyright: © Copyright 1978 American Mathematical Society