A commutant lifting theorem for a domain in \(\mathbb{C}^2\) and spectral interpolation
The version of the commutant lifting theorem which the authors extend is Ando's theorem: any commuting pair of contractions has a dilation consisting of a commuting pair of unitaries. Note that the joint spectrum of the pair of unitaries is the two dimensional torus which serves as the distinguished boundary of the bidisk. To state the main result, consider the set of complex pairs \(\Gamma_0=\{(a+b,ab):|a|\leq 1,|b|\leq 1\}\) and its closure \(\Gamma\). Let \((S,P)\) be a pair of commuting operators on a Hilbert space. Then the authors show that this pair has a commuting normal dilation \((S',T')\) with spectrum in the distinguished boundary of \(\Gamma\) if and only if \(\Gamma\) is a complete spectral set for \((S,T)\), and if and only if certain positivity conditions are satisfied. In addition, the pair \((S',T')\) can be chosen of the form \((U+V,UV)\), where \(U\) and \(V\) are commuting unitary operators. The main result is also applied to an interpolation problem for \(2\times 2\) matrix-valued functions in the unit disk, such that the eigenvalues of the matrices are always in the closed unit disk.
interpolation problem for \(2\times 2\) matrix-valued functions
scientific article; zbMATH DE number 1270855