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Spectrum of a composition operator

Author: William C. Ridge
Journal: Proc. Amer. Math. Soc. 37 (1973), 121-127
MSC: Primary 28A65; Secondary 47A10
MathSciNet review: 0306457
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Abstract: A composition operator is a linear operator induced on a subspace of $ {K^X}$ by a point transformation $ \phi $ on a set X (where K denotes the scalar field) by the formula $ Tf(x) = f \circ \phi (x)$. Familiar examples include translation operators on the real line and on topological groups, analytic functions which preserve the class of harmonic functions (and Green's functions), ergodic transformations which induce unitary operators on $ {L^2}$, shift and weighted shift operators.

The spectrum, approximate point spectrum, and point spectrum of an $ {L^p}$-composition operator have circular symmetry about 0, except on the unit circle, where they form unions of subgroups; certain consequences are derived from this.

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

  • [1] Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
  • [2] W. C. Ridge, Composition operators, Thesis, Indiana University, Bloomington, Ind., 1969, pp. 1-46.
  • [3] Eric A. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442–449. MR 0223914

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Article copyright: © Copyright 1973 American Mathematical Society