A Characterization of Resolvent Algebra For Some Operators

We characterize the resolvent subalgebra $R_A$ for a matrix operator $A$ on a finite dimensional Hilbert space $\mathcal{H}$. Then we show that $R_A = R_A^c=\big\{ T \in \mathcal{L}(\mathcal{H}) : \mathcal{N}(A)\in \mathrm{Lat}(T)\ \big\}$. Also in the infinite dimensional case we show that if $A$ is surjective or bounded bellow then $R_A=\big\{ T \in \mathcal{L}(\mathcal{H}) : \mathcal{N}(A)\in \mathrm{Lat}(T)\ \big\}$.