On the operator ranges of analytic functions

Abstract

Following Doob, we say that a function f ( z ) f(z) analytic in the unit disk U U has the property K ( ρ ) K(\rho ) if f ( 0 ) = 0 f(0) = 0 and for some arc γ \gamma on the unit circle whose measure | γ | ⩾ 2 ρ > 0 \left | \gamma \right | \geqslant 2\rho > 0 , \[ lim inf j → ∞ | f ( z j ) | ⩾ 1 where z j → z ∈ γ and z j ∈ U . \lim \inf \limits _{j \to \infty } \left | {f({z_j})} \right | \geqslant 1\quad {\text {where}}\;{z_j} \to z \in \gamma \;{\text {and}}\;{z_j} \in U. \] Let H H be a Hilbert space over the complex field, A A an operator whose spectrum is included in U U , | | A | | || A || the operator norm of A A , and f ( A ) f(A) the usual Riesz-Dunford operator. We prove that there is no function with the property K ( ρ ) K(\rho ) satisfying \[ ( 1 − | | A | | ) | | f ′ ( A ) | | ⩽ 1 / n for all | | A | | > 1 , (1 - || A ||)|| {f’(A)} || \leqslant 1/n\quad {\text {for}}\;{\text {all}}\;|| A || > 1, \] where n > N ( ρ ) = log ⁡ ( 1 / ( 1 − cos ⁡ ρ ) ) n > N(\rho ) = \log (1/(1 - \cos \rho )) . We also show that if f f has the property K ( ρ ) K(\rho ) then the operator range of f ( A ) f(A) covers a ball of radius k ( ρ ) = 3 / ( 4 N ( ρ ) ) 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 4 -fold univalent function is not convex. This extends our solution to Ky Fan’s Problem [4].