Extinction in lower Hessenberg branching processes with countably many types

We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton.Watson processes with typeset .={0,1,2,.} set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q, and whose maximum is the partial extinction probability vector q.. In the case where q.=1, we derive a global extinction criterion which holds under second moment conditions, and when q.<1, we develop necessary and sufficient conditions for q=q.. We also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector q..