Systems of operator equations and perturbation of spectral subspaces of commuting operators (Q1110787)

Let \(\{A_ j:1\leq j\leq m\}\) be a set of commuting continuous linear operators on a Banach space X and let \(\{B_ j:1\leq j\leq m\}\) be a set of commuting continuous linear operators on a Banach space Y. The authors prove the existence and uniqueness of a continuous linear operator Q:Y\(\to X\) satisfying the system of equations \(A_ jQ-QB_ j=U_ j\) (1\(\leq j\leq m)\), where \(U_ j:Y\to X\) (1\(\leq j\leq m)\) are given continuous linear operators. Estimates for the norm \(\| Q\|\) are also given. The results are applied to the perturbation of spectra and spectral subspaces of classes of commuting m-tuples of operators. The equation \[ \sum^{m}_{j=1}A_ jQB_ j=U\quad with\quad U\in L(Y,X) \] is also studied and the existence of a unique solution is proved and the norm \(\| Q\|\) is estimated.