Semigroups of linear operators on \(p\)-Fréchet spaces, \(0<p<1\)
A \(p\)-Fréchet space (sometimes also called a \(p\)-Banach space), \(0<p<1\), is a linear space \((X,+,\| \cdot\| )\) over \(K=\mathbb R\) or \(K=\mathbb C\) which satisfies \(\| x+y\| \leqslant \| x\| +\| y\| \) for all \(x,y\in K\), \(\| x\| =0\) if and only if \(x=0\), \(\| \lambda x\| =| \lambda| ^p\| x\| \) for all \(\lambda \in K\), \(x\in X\), and is complete with respect to the metric \(d(x,y)=\| x-y\| \). Such a space is not necessarily locally compact: consider the Hardy space \(H^p\), the sequence space \(\ell^p\), and the spaces \(L^p\), all with \(0<p<1\). If \(A:X\to X\) is linear and continuous, then \((\sum^m_{j=0}(t^j/j!)A^j(x))_{m=1,2,\dots}\), \(t\in \mathbb R\), \(x\in X\), is a Cauchy sequence, the limit of which defines a semigroup. The authors develop the foundations of a theory of such semigroups and discuss several applications.
scientific article; zbMATH DE number 5150249