Pseudo $S$-spectra of special operators in quaternionic Hilbert spaces

For a bounded quaternionic operator $T$ on a right quaternionic Hilbert space $\mathcal{H}$ and $\epsilon >0$, the pseudo $S$-spectrum of $T$ is defined as \begin{align*} Λ_{\varepsilon}^{S}(T) := σ_S (T) \bigcup \left { q \in $\mathbb{H}$\setminus σ_S(T):; |Δ_{q}(T)^{-1}| \geq \frac{1}{\varepsilon} \right}, \end{align*} where $\mathbb{H}$ denotes the division ring of quaternions, ${\sigma }_S(T)$ is the $S$-spectrum of $T$ and ${\mathrm{\Delta }}_q(T)=T^2-2\text{Re}(q)T+\mathrm{\mid }q{\mathrm{\mid }}^{2}I$. This is a natural generalization of pseudospectrum from the theory of complex Hilbert spaces. In this article, we investigate several properties of the pseudo $S$-spectrum and explicitly compute the pseudo $S$-spectra for some special classes of operators such as upper triangular matrices, self adjoint-operators, normal operators and orthogonal projections. In particular, by an application of $S$-functional calculus, we show that a quaternionic operator is a left multiplication operator induced by a real number $r$ if and only if for every $\epsilon >0$ the pseudo $S$-spectrum of the operator is the circularization of a closed disc in the complex plane centered at $r$ with the radius $\sqrt{\epsilon }$. Further, we propose a $G_1$-condition for quaternionic operators and prove some results in this setting.