Composition operators with multivalent symbol
Abstract
This thesis contains two results on a class of composition operators with multivalent symbol. Both of the results concern composition operators $C\sb\varphi$ where $\varphi(z) = {{(1-2c)z\sp2}\over{1-2cz}}$ for $0< c < 1/2$. The result in Chapter 2 is a complete and convenient description of the kernel of the adjoint $C\sbsp{\varphi}{\*}$. In Chapter 3 we identify a subspace of $H\sp2(D)$ on which $C\sbsp{\varphi}{\*}$ behaves like a weighted shift. This reproduces the description of the spectrum found in (6) and shows that for this class of composition operators that the interior of the spectrum is a disk of eigenvalues of $C\sbsp{\varphi}{\*}$ of infinite multiplicity. The techniques of proof used are similar to those of (2). Chapter 4 contains a calculation of the formal adjoint for this class of composition operators and provides the intuition for the result in Chapter 2. This calculation is only formal since the operators involved are not defined on $H\sp2(D).$
Degree
Ph.D.
Advisors
Cowen, Purdue University.
Subject Area
Mathematics
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