Resolvents of operators on tensor products of Euclidean spaces

We consider the operator T = m k=1 A 1k ⊗ A 2k (1 ≤ m < ∞), where A lk are n l × n l matrices (k = 1,. .. , m; l = 1, 2), ⊗ means the tensor product. Norm estimates for the resolvent of that operator are derived. By these estimates, we obtain bounds for a solution X of the equation m k=1 A 1k X A 2k = C and explore perturbations of that equation. The norm estimates for the resolvent of T enable us to establish a bound for the distance between invariant subspaces of two matrices. Besides, the well-known Davis–Kahan result is particularly generalized. In addition, we derive a new stability test for non-linear non-autonomous ordinary differential equations.