On the Operator Ranges of Analytic Functions

J. S. Hwang
1983 Proceedings of the American Mathematical Society  
Following Doob, we say that a function/(z) analytic in the unit disk U has the property K(p) if/(0) = 0 and for some arc y on the unit circle whose measure | y \ > 2p > 0, liminf \f(z])\> 1 where z -: G y and z, E U. J-X Let H be a Hubert space over the complex field, A an operator whose spectrum is included in U, IIAW the operator norm of A. and f(A) the usual Riesz-Dunford operator. We prove that there is no function with the property K( p) satisfying (1 -\\A\\)\\f'(A)\\ « \/n for all IM || <
more » ... \/n for all IM || < 1. where n > N( p ) = log( 1/(1-cos p )). We also show that if / has the property K( p ) then the operator range of f(A) covers a ball of radius k(p) = ]/3 /(4N(p)). 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].
doi:10.2307/2044359 fatcat:2jkwydrgbbg7ve2uptz6y4vwfi