On the operator ranges of analytic functions

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 $\left | \gamma \right | \geqslant 2\rho > 0$, \[ \lim \inf \limits _{j \to \infty } \left | {f({z_j})} \right | \geqslant 1\quad {\text {where}}\;{z_j} \to z \in \gamma \;{\text {and}}\;{z_j} \in U.\] Let $H$ be a Hilbert space over the complex field, $A$ an operator whose spectrum is included in $U$, $|| A ||$ 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 \[ (1 - || A ||)|| {f’(A)} || \leqslant 1/n\quad {\text {for}}\;{\text {all}}\;|| A || < 1,\] 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].