Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 1-9, 2017

On invariant subspaces for polynomially bounded operators

Junfeng Liu

Received October 3, 2014.   First published February 24, 2017.

Junfeng Liu, Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau 999078, P. R. China, e-mail: jfliu997@163.com

Abstract: We discuss the invariant subspace problem of polynomially bounded operators on a Banach space and obtain an invariant subspace theorem for polynomially bounded operators. At the same time, we state two open problems, which are relative propositions of this invariant subspace theorem. By means of the two relative propositions (if they are true), together with the result of this paper and the result of C. Ambrozie and V. Müller (2004) one can obtain an important conclusion that every polynomially bounded operator on a Banach space whose spectrum contains the unit circle has a nontrivial invariant closed subspace. This conclusion can generalize remarkably the famous result that every contraction on a Hilbert space whose spectrum contains the unit circle has a nontrivial invariant closed subspace (1988 and 1997).

Keywords: polynomially bounded operator; invariant subspace

Classification (MSC 2010): 47A15

DOI: 10.21136/CMJ.2017.0459-14

Full text available as PDF.


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