A1 Refereed original research article in a scientific journal
Rigidity of composition operators on the Hardy space H-P
Authors: Laitila J, Nieminen PJ, Saksman E, Tylli HO
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
Publication year: 2017
Journal: Advances in Mathematics
Journal name in source: ADVANCES IN MATHEMATICS
Journal acronym: ADV MATH
Volume: 319
First page : 610
Last page: 629
Number of pages: 20
ISSN: 0001-8708
eISSN: 1090-2082
DOI: https://doi.org/10.1016/j.aim.2017.08.029(external)
Abstract
Let phi be an analytic map taking the unit disk ID into itself. We establish that the class of composition operators f bar right arrow C-phi(f) = f o phi exhibits a rather strong rigidity of non-compact behaviour on the Hardy space H-P, for 1 <= p < infinity and p not equal 2. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) C-phi is a compact operator H-P -> H-P, (ii) C-phi fixes a (linearly isomorphic) copy of l(P) in H-P, but C-phi does not fix any copies of l(2) in H-P, (iii) C-phi fixes a copy of l(2) in H-P. Moreover, in case (iii) the operator C-phi actually fixes a copy of L-P(0, 1) in H-P provided p > 1. We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on H-P, which contain the compact operators k(H-P). In particular, the class of composition operators on H-P does not reflect the quite complicated lattice structure of such ideals. (C) 2017 Elsevier Inc. All rights reserved.
Let phi be an analytic map taking the unit disk ID into itself. We establish that the class of composition operators f bar right arrow C-phi(f) = f o phi exhibits a rather strong rigidity of non-compact behaviour on the Hardy space H-P, for 1 <= p < infinity and p not equal 2. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) C-phi is a compact operator H-P -> H-P, (ii) C-phi fixes a (linearly isomorphic) copy of l(P) in H-P, but C-phi does not fix any copies of l(2) in H-P, (iii) C-phi fixes a copy of l(2) in H-P. Moreover, in case (iii) the operator C-phi actually fixes a copy of L-P(0, 1) in H-P provided p > 1. We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on H-P, which contain the compact operators k(H-P). In particular, the class of composition operators on H-P does not reflect the quite complicated lattice structure of such ideals. (C) 2017 Elsevier Inc. All rights reserved.
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