On $n$-parameter discrete and continuous semigroups of operators

Abstract:

We prove that n commuting operators on a Hilbert space can be uniquely simultaneously extended to doubly commuting coisometric operators if and only if they satisfy certain positivity conditions, which for the case $n = 1$ state simply that the original operator is a contraction. Our proof establishes the connection between these positivity conditions and the backward translation semigroup on ${l^2}({Z^{ + n}},\mathcal {K})$. A semigroup of operators is unitarily equivalent to backward translation (or a part thereof) on ${l^2}({Z^{ + n}},\mathcal {K})$ if and only if the positivity conditions are satisfied and the individual operators are coisometries (or contractions) whose powers tend strongly to zero. Analogous results are proven in the continuous case ${R^{ + n}}$.