Saturation of iterations of bounded linear operators
Let \({\mathcal X}\) be a Banach space and \({\mathcal B}({\mathcal X})\) the algebra of all bounded linear operators on \({\mathcal X}\). Let \(\{L_{\alpha}:\) \(\alpha\in D\}\) be an approximation process on \({\mathcal X}\), i.e. a net in \({\mathcal B}({\mathcal X})\) converging strongly to the identity operator I on \({\mathcal X}\), and let \(\{x_{\alpha}:\) \(\alpha\in D\}\) be a net of positive numbers converging to zero. Define the trivial class of \(\{L_{\alpha}\}\) to be \[ T[{\mathcal X};\quad \{L_{\alpha}\}]=\{x\in {\mathcal X}:\quad L_{\alpha}x=x\quad for\quad all\quad \alpha \in D\} \] and let \[ S[{\mathcal X};\quad \{L_{\alpha}\},\{x_{\alpha}\}]=\{x\in {\mathcal X}:\quad \| L_{\alpha}x-x\| =O(x_{\alpha})\}. \] If \(\| L_{\alpha}x-x\| =o(x_{\alpha})\) implies that \(f\in T[{\mathcal X}\); \(\{L_{\alpha}\}]\) and if \(S[{\mathcal X};\quad \{L_{\alpha}\},\{x_{\alpha}\}]\setminus T[{\mathcal X};\quad \{L_{\alpha}\}]\neq \emptyset,\) then it is said that \(\{L_{\alpha}\}\) is saturated in \({\mathcal X}\). In this case \(S[{\mathcal X};\quad \{L_ a\},\{x_{\alpha}\}]\) is called the saturation class of \(\{L_{\alpha}\}.\) In this paper the author continues his previous work and studies the existence of the saturation order \(\{x_{\alpha}\}\) and characterization of the saturation class of a net \(\{L^ k_{\alpha}:\) \(\alpha\in D\}\) of the kth iterations of \(L_{\alpha}\), \(\alpha\in D\), where k is an arbitrary fixed positive integer. Then the results are applied to approximation processes generated by semigroups of operators on \({\mathcal X}\).
scientific article; zbMATH DE number 4140605