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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the operator ranges of analytic functions

Author: J. S. Hwang
Journal: Proc. Amer. Math. Soc. 87 (1983), 90-94
MSC: Primary 47A60; Secondary 30C45
MathSciNet review: 677239
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Abstract: Following Doob, we say that a function $ f(z)$ analytic in the unit disk $ U$ has the property $ K(\rho )$ if $ f(0) = 0$ and for some arc $ \gamma $ on the unit circle whose measure $ \left\vert \gamma \right\vert \geqslant 2\rho > 0$,

$\displaystyle \mathop {\lim \inf }\limits_{j \to \infty } \left\vert {f({z_j})}... ...lant 1\quad {\text{where}}\;{z_j} \to z \in \gamma \;{\text{and}}\;{z_j} \in U.$

Let $ H$ be a Hilbert space over the complex field, $ A$ an operator whose spectrum is included in $ U$, $ \vert\vert A \vert\vert$ the operator norm of $ A$, and $ f(A)$ the usual Riesz-Dunford operator. We prove that there is no function with the property $ K(\rho )$ satisfying

$\displaystyle (1 - \vert\vert A \vert\vert)\vert\vert {f'(A)} \vert\vert \leqslant 1/n\quad {\text{for}}\;{\text{all}}\;\vert\vert A \vert\vert < 1,$

where $ n > N(\rho ) = \log (1/(1 - \cos \rho ))$. We also show that if $ f$ has the property $ K(\rho )$ then the operator range of $ f(A)$ covers a ball of radius $ k(\rho ) = \sqrt 3 /(4N(\rho ))$. These two results generalize our previous solutions of two long open problems of Doob [1]. Finally, we prove that the operator range of any $ 4$-fold univalent function is not convex. This extends our solution to Ky Fan's Problem [4].

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Keywords: Analytic function, Riesz-Dunford operator, convex function and operator range
Article copyright: © Copyright 1983 American Mathematical Society