caa a22 4500 467931801 CHVBK 20180406152945.0 cr unu---uuuuu 170328e20061101xx s 000 0 eng 10.1007/s00020-006-1420-x doi (NATIONALLICENCE)springer-10.1007/s00020-006-1420-x Composition Operators on Small Spaces [Elektronische Daten] [Boo Choe, Hyungwoon Koo, Wayne Smith] Abstract.: We show that if a small holomorphic Sobolev space on the unit disk is not just small but very small, then a trivial necessary condition is also sufficient for a composition operator to be bounded. A similar result for holomorphic Lipschitz spaces is also obtained. These results may be viewed as boundedness analogues of Shapiro's theorem concerning compact composition operators on small spaces. We also prove the converse of Shapiro's theorem if the symbol function is already contained in the space under consideration. In the course of the proofs we characterize the bounded composition operators on the Zygmund class. Also, as a by-product of our arguments, we show that small holomorphic Sobolev spaces are algebras. Birkhäuser Verlag, Basel, 2006 Composition operator nationallicence fractional derivative nationallicence small space nationallicence Choe Boo Department of Mathematics, Korea University, 136-701, Seoul, Korea aut Koo Hyungwoon Department of Mathematics, Korea University, 136-701, Seoul, Korea aut Smith Wayne Department of Mathematics, University of Hawaii, 96822, Honolulu, Hawaii, USA aut Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhäuser.ch 56/3(2006-11-01), 357-380 0378-620X 56:3<357 2006 56 20 https://doi.org/10.1007/s00020-006-1420-x text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00020-006-1420-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Choe Boo Department of Mathematics, Korea University, 136-701, Seoul, Korea aut NATIONALLICENCE 700 1- Koo Hyungwoon Department of Mathematics, Korea University, 136-701, Seoul, Korea aut NATIONALLICENCE 700 1- Smith Wayne Department of Mathematics, University of Hawaii, 96822, Honolulu, Hawaii, USA aut NATIONALLICENCE 773 0- Integral Equations and Operator Theory Birkhäuser-Verlag; www.birkhäuser.ch 56/3(2006-11-01), 357-380 0378-620X 56:3<357 2006 56 20 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer