On the Relation Between S-Spectrum and Right Spectrum

We use the \(\mathbb {R}\) $\mathbb{R}$ -linearity of \(I\lambda -T\) $I\lambda -T$ to define \(\sigma (T)\) $\sigma (T)$ , the right spectrum of a right \(\mathbb {H}\) $\mathbb{H}$ -linear operator T in a right quaternionic Banach space. We show that \(\sigma (T)\) $\sigma (T)$ coincides with the S-spectrum \(\sigma _S(T)\) ${\sigma }_S(T)$ .