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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] Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, With the assistance of William G. Bade and Robert G. Bartle, Interscience Publishers John Wiley & Sons New York-London, 1963. MR 0188745
  • [3] Igor Kluvánek and Greg Knowles, Vector measures and control systems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. North-Holland Mathematics Studies, Vol. 20; Notas de Matemática, No. 58. [Notes on Mathematics, No. 58]. MR 0499068

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Keywords: Scalar operator, vector measures
Article copyright: © Copyright 1978 American Mathematical Society