A Note on the Spectral Mapping Theorem
Tetsuro Yamamoto · SIAM Journal on Mathematical Analysis · 1971
Let A be a linear operator on a complex normed linear space (into itself). We denote by $N(A)$ and $R(A)$ the null-space and the range of A, respectively. The purpose of this paper is to show that \[ N\left( {\prod_{i = 1}^s {\left( {A - \alpha _i I} \right)^{n_i } } } \right) = \sum _{i = 1}^s { \oplus N\left( {\left( {A - \alpha _i I} \right)^{n_i } } \right)\quad ({\text{direct sum}})} \] and \[R\left( {\prod_{i = 1}^s {\left( {A - \alpha _i I} \right)^{n_i } } } \right) = \bigcap\limits_{i = 1}^s {R\left( {\left( {A - \alpha _i I} \right)^{n_i } } \right),} \] where the $\alpha _i $ are different complex numbers and I is the identity operator, and to note that some theorems in spectral theory follow directly from these formulas.