Journal of Operator Theory
Volume 35, Issue 1, Winter 1996 pp. 107-115.
A parametrization of canonically Koszul invertible pairsAuthors: Ximena P. Catepillán
Author institution:Department of Mathematics, Millersville University of Pennsylvania, Millersville, PA 17551, U.S.A., e-mail: xcatepil@marauder.millersv.edu
Summary: Let $T = (T_1, T_2)$ be a commuting pair of operators on a Hilbert space $\mathcal H$, and let $T_i = V_iP_i, i = 1, 2,$ be the polar decompositions of $T_1$ and $T_2$. The pair $T$ is called canonically Koszul invertible if the Koszul complex $K(T, \mathcal H)$ admits a $C^*$-split, i.e., if $[(D^0)^*]D^0]^{-1}(D^0)^*$ and $(D^1 )^*[D^1 (D^1 )^*]^{-1}$ are the boundary maps of a Koszul complex, where $D^0$ and $D^1$ are the boundaries of $K(T, \mathcal H)$. We find a parametrization of canonically Koszul invertible pairs in terms of the factors $V_1, P_1, V_2$ and $P_2$. In addition, we obtain a new characterization of the commutant spectrum of $T$.
Keywords: Operators on Hilbert space, Koszul complex, commutant spectrum.
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