Abstract
Abstract In this paper we investigate the numerical range of C* bφ m Caφ n and Caφ n C* bφ m on the Hardy space where φ is an inner function fixing the origin and a and b are points in the open unit disc. In the case when |a| = |b| = 1 we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.
Highlights
In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace
The Hardy space, H (D), consists of all holomorphic functions f on the open unit disc D whose radial means are uniformly bounded in L (T)
The Hardy space is the domain for the class of composition operators Cφ de ned by
Summary
Hardy space where φ is an inner function xing the origin and a and b are points in the open unit disc. |a| = |b| = we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace
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