Resolvents of operators on tensor products of Euclidean spaces

We consider the operator (Formula presented.) where (Formula presented.) are (Formula presented.) matrices (Formula presented.) , (Formula presented.) means the tensor product. Norm estimates for the resolvent of that operator are derived. By these estimates, we obtain bounds for a solution (Formula presented.) of the equation (Formula presented.) and explore perturbations of that equation. The norm estimates for the resolvent of (Formula presented.) 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.