Absolute equivalence and Dirac operators of commuting tuples of operators (Q1772909)

The authors introduce an equivalence relation of bounded linear operators acting in pairs of Hilbert spaces, which they call absolute equivalence. Two operators are called absolutely equivalent if both the absolute values of the operators and the absolute values of their adjoints are pairwise equivalent. In the case of finite-dimensional Hilbert spaces, operators (matrices) are absolutely equivalent precisely when they have the same singular value decomposition. Absolute equivalence is then used in the paper to study the Koszul complex and the Dirac operator (as defined by \textit{W. Arveson} [J. Funct. Anal. 189, No.~1, 53--79 (2002; Zbl 1038.47003)]) of a commutative tuple of operators. In particular, the Dirac operators of an invertible, Fredholm, normal, Hermitian, spherical unitary tuple of commuting operators are characterized.