Quaternionic spherical sectorial and dissipative operators via S-resolvent kernels

In this paper, we treat the quaternionic functional calculus for right linear quaternionic operators whose components do not necessarily commute and develop a theory of quaternionic non-negative operator, spherical sectorial operator and dissipative operator via S-resolvent kernels in quaternionic locally convex spaces (short for $q.l.c.s$.). The notions of quaternionic non-negative operators and quaternionic (m-)dissipative operators are introduced via S-resolvent operators and $H$-valued inner product on Hilbert $H$-bimodule. By choosing the suitable spherical sector, the spherical sectorial operator is introduced to establish the relationship with the quaternionic non-negative operator. It is crucial to note that the quaternionic operators we consider do not necessarily commute.