TY - JOUR
T1 - Composition operators on uniform algebras and the pseudohyperbolic metric
AU - Galindo, P.
AU - Gamelin, T. W.
AU - Lindström, Mikael
PY - 2004
Y1 - 2004
N2 - Let A be a uniform algebra, and let φ be a self-map of the spectrum MA of A that induces a composition operator Cφ on A. It is shown that the image of MA under some iterate φn of φ is hyperbolically bounded if and only if φ has a finite number of attracting cycles to which the iterates of φ converge. On the other hand, the image of the spectrum of A under φ is not hyperbolically bounded if and only if there is a subspace of A** "almost" isometric to ℓ∞ on which Cφ** is "almost" a,n isometry. A corollary of these characterizations is that if Cφ is weakly compact, and if the spectrum of A is connected, then φ has a unique fixed point, to which the iterates of φ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].
AB - Let A be a uniform algebra, and let φ be a self-map of the spectrum MA of A that induces a composition operator Cφ on A. It is shown that the image of MA under some iterate φn of φ is hyperbolically bounded if and only if φ has a finite number of attracting cycles to which the iterates of φ converge. On the other hand, the image of the spectrum of A under φ is not hyperbolically bounded if and only if there is a subspace of A** "almost" isometric to ℓ∞ on which Cφ** is "almost" a,n isometry. A corollary of these characterizations is that if Cφ is weakly compact, and if the spectrum of A is connected, then φ has a unique fixed point, to which the iterates of φ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].
KW - uniform algebra
KW - composition operator
KW - hyperbolically bounded
KW - interpolating sequence
UR - https://www.mendeley.com/catalogue/23e3ba51-0e4b-3ddc-98c4-5eb42106a1dc/
U2 - 10.4134/JKMS.2004.41.1.001
DO - 10.4134/JKMS.2004.41.1.001
M3 - Article
SN - 0304-9914
VL - 41
SP - 1
EP - 20
JO - Journal- Korean Mathematical Society
JF - Journal- Korean Mathematical Society
IS - 1
ER -