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Adjoint Abelian Operators on Lp and C(K)

Richard J. Fleming and James E. Jamison
Transactions of the American Mathematical Society
Vol. 217 (Mar., 1976), pp. 87-98
DOI: 10.2307/1997559
Stable URL: http://www.jstor.org/stable/1997559
Page Count: 12
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Adjoint Abelian Operators on Lp and C(K)
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Abstract

An operator A on a Banach space X is said to be adjoint abelian if there is a semi-inner product [ ·,· ] consistent with the norm on X such that [ Ax, y ] = [ x, Ay ] for all x, y ∈ X. In this paper we show that every adjoint abelian operator on C(K) or $L^p(\Omega, \Sigma, \mu) (1 < p < \infty, p \neq 2)$ is a multiple of an isometry whose square is the identity and hence is of the form Ax(·) = λ α(·)(x ⚬ φ)(·) where α is a scalar valued function with α(·)α ⚬ φ(·) = 1 and φ is a homeomorphism of K (or a set isomorphism in case of Lp(Ω, Σ, μ)) with $\phi \circ \phi = \operatorname{identity}$ (essentially).

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