The set of periods of periodic points of a linear operator
For each \(n \in {\mathbb N}\) define the set \[ F_n=\{{\{1\} \cup \tilde A:A \subset {\mathbb N}\text{ and }|A|\leq n/2} \}\cup \{{\{1\}\cup \tilde A:A \subset {\mathbb N}\backslash \{1\},\;2 \in A\text{ and }|A|= (n+1)/2}\}, \] where \(| A |\) denotes the cardinality of \(A\) and \(\tilde A\) denotes the smallest subset of \({\mathbb N}\) containing \(A\) and closed under the binary operation least common multiple. Note also that \(F_{2m}= \{\{1\}\cup \tilde A:A \subset{\mathbb N}\) and \(|A|\leq m\}\) for all \(m \in {\mathbb N}\). The main results of this note are the following: (i) \(F_{2n}\) is the family of period sets of linear operators on \({\mathbb C}\); (ii) \(F_n\) is the family of period sets of linear operators on \({\mathbb R}\); (iii) \(F = \bigcup_n F_n\) is the family of period sets of linear operators on the Hilbert space \(l^2\) having finite rank (finite dimensional range); (iv) \(\{ {A \subset {\mathbb N}:A = \tilde A\text{ and }1 \in A} \}\) is the family of period sets of isometries on \(l^2\).
scientific article; zbMATH DE number 5561624