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# The Solution Sets of Extremal Problems in H1

Eric Hayashi
Proceedings of the American Mathematical Society
Vol. 93, No. 4 (Apr., 1985), pp. 690-696
DOI: 10.2307/2045546
Stable URL: http://www.jstor.org/stable/2045546
Page Count: 7
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## Abstract

Let u be an essentially bounded function on the unit circle T. Let Su denote the subset of the unit sphere of H1 on which the functional $F \mapsto \int^{2\pi}_0 \bar{u}(e^{it})F(e^{it}) dt/2\pi$ attains its norm. A complete description of Su is given in terms of an inner function b0 and an outer function g0 in H2 for which g2 0 is an exposed point in the unit ball of H1. An explicit description is given for the kernel of an arbitrary Toeplitz operator on H2. The exposed points in H1 are characterized; an example is given of a strong outer function in H1 which is not exposed.

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