Null-orbit reflexive operators (Q2914874)

The authors introduce and study the concept of null-orbit reflexivity for operators on Banach spaces. A bounded linear operator \(T\) on a Banach space \(X\) is said to be null-orbit reflexive if the following property is satisfied: if \(S\) is a bounded linear operator on \(X\) such that \(Sx\) belongs to the closure of the set \(\{0\} \cup \{T^nx : n \geq 0\}\) for every \(x \in X\), then \(S\) belongs to the closure of the set \(\{0\} \cup \{T^n : n \geq 0\}\) with respect to the strong operator topology. This concept is closely related to the notion of orbit reflexive operators (cf.\ \textit{D. Hadwin} et al. [J. Lond.\ Math.\ Soc., II. Ser.\ 34, 111--119 (1986; Zbl 0624.47002)]) and also to the recent notion of \(\mathbb{C}\)-orbit reflexive operators (cf.\ \textit{D. Hadwin} et al. [Oper.\ Matrices 5, No.\ 3, 511--527 (2011; Zbl 1233.47002)]).NEWLINENEWLINEMany sufficient conditions for null-orbit reflexivity are presented and many classes of operators on Banach and Hilbert spaces are shown to be null-orbit reflexive, including hyponormal, algebraic, compact, strictly block-upper (lower) triangular operators, and operators whose spectral radius is not \(1\). Moreover, every polynomially bounded Hilbert space operator is shown to be both orbit reflexive and null-orbit reflexive.NEWLINENEWLINEThe first example of a Hilbert space operator that is not orbit reflexive was given by \textit{S. Grivaux} and \textit{M. Roginskaya} [Int.\ Math.\ Res.\ Not.\ 2008, Article ID rnn083 (2008; Zbl 1159.47003)]. A much simpler example of such an operator was obtained later by \textit{V. Müller} and \textit{J. Vršovský} [J. Lond.\ Math.\ Soc., II. Ser.\ 79, No.\ 2, 497--510 (2009, Zbl 1168.47006)]. It is shown that the second of these examples is null-orbit reflexive, but the first one is not.NEWLINENEWLINEThe authors also introduce and study an algebraic notion of null-orbit reflexivity for linear operators on a linear space.