Abstract. A remarkable theorem of Domar asserts that the lattice of the invariant subspaces of the right shift semigroup $\{S_{\tau }\}_{\tau\geq 0}$ in $L^2({\msbm R}R _+, w(t)dt)$ consists of just the \textit{``standard invariant subspaces''} whenever $w$ is a positive continuous function in ${\msbm R}R _+$ such that (1) $\log w$ is concave in $[c,\infty )$ for some $c\geq0$, (2) $\lim_ {t\to\infty }\frac{-\log w(t)}{t}=\infty,$ and $\lim_ {t\to\infty }\frac{\log |\log w(t)|-\log t}{\sqrt{\log t}}=\infty.$ We prove an extension of Domar's Theorem to a strictly wider class of weights $w$, answering a question posed by Domar in [Do3].

DOI: 10.14232/actasm-015-837-7

AMS Subject Classification (1991): 47A15

Keyword(s): right-translation invariant subspaces

Received December 17, 2015. (Registered under 87/2015.)