In this paper, we present and study the ascent S-spectrum (ASS) and descent S-spectrum (DSS) of a bounded right linear operator T defined on a two-sided quaternionic Banach space X, as well as, the essential ascent S-spectrum (EASS) and essential descent S-spectrum (EDSS). On the one hand, we establish a connection between ASS, DSS, EASS, and EDSS with the usual ascent, descent, essential ascent, and essential descent spectra, respectively. Furthermore, we show that, under certain conditions, the spectral mapping theorem holds for these spectra for any intrinsic slice hyperholomorphic function f defined on \(\sigma _S(T).\) On the other hand, we prove some analogous properties known in the complex framework for their quaternionic counterparts. In particular, if s is an arbitrary element of the essential ascent S-spectrum (respectively, the essential descent S-spectrum), then the pseudo-resolvent \(Q_q(T)\) of T is semi-regular and upper semi-Fredholm (respectively, semi-regular and lower semi-Fredholm) for all q in \(\mathcal {V}\setminus [s],\) where [s] is the 2-dimensional sphere associated to s and \(\mathcal {V}\) is a neighborhood of s. As a result, the ASS, DSS, EASS, and EDSS are compact subsets of the S-spectrum. Moreover, this work introduces the concept of spherical poles of the S-resolvent, the quaternionic analog of the resolvent poles in the complex frame. We investigate the relationships between these poles and the ascent/descent S-spectra (ASS/DSS) and their essential counterparts (EASS/EDSS). Finally, we establish that the ASS, DSS, and EASS of the product operator TS (excluding zero) coincide with the corresponding spectra of ST (excluding zero).
