A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
Rigidity of composition operators on the Hardy space H-P
Tekijät: Laitila J, Nieminen PJ, Saksman E, Tylli HO
Kustantaja: ACADEMIC PRESS INC ELSEVIER SCIENCE
Julkaisuvuosi: 2017
Journal: Advances in Mathematics
Tietokannassa oleva lehden nimi: ADVANCES IN MATHEMATICS
Lehden akronyymi: ADV MATH
Vuosikerta: 319
Aloitussivu: 610
Lopetussivu: 629
Sivujen määrä: 20
ISSN: 0001-8708
eISSN: 1090-2082
DOI: https://doi.org/10.1016/j.aim.2017.08.029
Tiivistelmä
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.
Turun yliopiston Tutkimustietojärjestelmän portaali