Junfeng Liu, Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau 999078, P. R. China, e-mail: firstname.lastname@example.org
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
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