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Normal weighted composition operators on the Hardy space | | Paul S. Bourdon
; Sivaram K. Narayan
; | | Date: |
7 Oct 2009 | | Abstract: | Let g be an analytic function on the open unit disc U such that g(U) is
contained in U, and let h be an analytic function on U such that the weighted
composition operator W_{h,g) defined by W_{h,g}f = h f(g) is bounded on the
Hardy space H^2. We characterize those weighted composition operators on H^2
that are unitary, showing that in contrast to the unweighted case (h=1), every
automorphism of U induces a unitary weighted composition operator. A
conjugation argument, using these unitary operators, allows us to describe all
normal weighted composition operators on H^2 for which the inducing map g fixes
a point in U. This description shows both h and g must be linear fractional in
order for W_{h,g} to be normal (assuming g fixes a point in U). In general, we
show that if W_{h, g} is normal on H^2 and h is not the zero function, then g
must be either univalent on U or constant. Descriptions of spectra are provided
for the operator W_{h,g} when it is unitary or when it is normal and g fixes a
point in U. | | Source: | arXiv, 0910.1259 | | Services: | Forum | Review | PDF | Favorites | |
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