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

VL - 41

SP - 1

EP - 20

JO - Journal- Korean Mathematical Society

JF - Journal- Korean Mathematical Society

SN - 0304-9914

IS - 1

ER -