@article{eb679cbda3c24d75874688bbe02d6bde,

title = "Connected components in the space of composition operators on H∞ functions of many variables",

abstract = "Let E be a complex Banach space with open unit ball BE. The structure of the space of composition operators on the Banach algebra H∞(BE), of bounded analytic functions on BE with the uniform topology, is studied. We prove that the composition operators arising from mappings whose range lies strictly inside BE form a path connected component. When E is a Hilbert space or a C0(X)-space, the path connected components are shown to be the open balls of radius 2.",

author = "Richard Aron and Pablo Galindo and Mikael Lindstr{\"o}m",

note = "Funding Information: Let E denote a complex Banach space with open unit ball BE and let r : BE --~ BE be an analytic mapping, where F is also a complex Banach space. We will consider composition operators Cr defined by Cr = f o r acting from the uniform algebra H~C(BF) of all bounded analytic functions on BF into HO~ Our object of study is C(H~176 HC~ the space of all composition operators endowed with the operator norm topology. Motivated by earlier research of Shapiro and Sundberg \[19\]o n the space of composition operators on the Hardy space H 2, MacCluer, Ohno and Zhao \[15\c] haracterize the connected components and the isolated points in the space C(H~ 17H6~ ). Their work was extended in \[10\]t o the space of all endomorphisms of H ~17I6n the H ~ setting, the main result is that the (path) connected components are the open balls of radius 2. A key tool to obtain these results is the use of the so-called pseudohyperbolicd istance in the ball BE, which is just the Poincar5 pseudodistance in the case of the unit disc. There is no known (to the authors) formula for the pseudohyperbolic metric in general Banach spaces. Nevertheless, for some special Banach spaces such as Hilbert space or a Co(X) space, an explicit formula is known. Therefore, in these cases, we can prove that the open balls of radius 2 are the path connected components in C(H~176 H~176 and also that the composition operators arising from biholomorphic mappings are isolated points, a result which holds for a wide class of Banach spaces. In \[15\]i t is shown that in the H ~17c6a se, the set of composition operators which differ *The research of this author was supported by grant number SAB1999-0214 from the Ministcrio de Educacidn, Cultura y Deporte during his stay at the Universidad de Valencia. *The research of this author was partially supported DGES(Spaln) pr. 96-0758. *The research of this author was partially supported by Magnus Ehrnrooths stiftelse.",

year = "2003",

doi = "10.1007/BF02789591",

language = "English",

volume = "45",

pages = "1--14",

journal = "Integral Equations and Operator Theory",

issn = "0378-620X",

publisher = "Springer",

number = "1",

}