Bourdon and Shapiro conjecture
Subject Areas : Analyze
1 - گروه ریاضی، دانشکده علوم پایه، دانشگاه یاسوج، یاسوج، ایران
Keywords: Hardy space, Composition operator, Elliptic automorphism, Numerical range.,
Abstract :
The numerical range of composition operators on Hardy space whose symbol is an ellipse of finite order is not disk. This is the conjecture that Bourdon and Shapiro made in 2000, the International Year of Mathematics. Attempts have been made to prove or disprove it over the past few years, but a complete answer has not yet been obtained. Bourdon and Shapiro have considered the question of when the numerical range of a composition operator is a disk centered at the origin and have shown that this happens whenever the inducing map is a non-elliptic conformal automorphism of the unit disk. They also have shown that the numerical range of elliptic automorphism with order 2 is an ellipse with focus at ±1. Recently, Patton et al. Proved the problem for each finite linear operator of order 3. Patton, in particular, showed that the numerical range of composition operators on Hardy space with a minimum polynomial of z ^ 3-1 is not a disk. In this study, we prove that this conjecture is correct for a large family of such operators.
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