Constructions of semigroups of operators given norms
Let \(X\) be a normed space, \(G\) a semigroup, and \(T: G\to L(X)\) a representation of \(G\) in the algebra of bounded linear operators \(X\to X\) with the induced norm; i.e., \(T(g_1g_2)= T(g_1) T(g_2)\) for all \(g_1,g_2\in G\). Definition. We say that the representation \(T: G\to L(X)\) realizes a function \(N:G\to \mathbb{R}^+\) \((\mathbb{R}^+= [0,\infty))\) if \(\|T(g)\|_{L(X)}= N(g)\) for all \(g\in G\). If a function \(N: G\to\mathbb{R}^+\) is realized by some representation \(T: G\to L(X)\) then the former is submultiplicative; i.e., \(N(g_1g_2)\leq N(g_1)N(g_2)\) for all \(g_1,g_2\in G\). The author studies the questions, what a space \(X\) should be in order to admit a realization of a submultiplicative function \(N: G\to \mathbb{R}^+\) and which submultiplicative functions \(N: G\to\mathbb{R}^+\) can be realized by some representation \(T: G\to L(X)\) and for which semigroups \(G\) this is possible.
scientific article; zbMATH DE number 923508