Pseudospectra in Banach Jordan Algebras

The primary focus of this research is to broaden the concept of pseudo spectrum from operators or matrices to elements in a unital com- plex Banach Jordan algebra-transcending from the associative to the non- associative setting. We introduce the notion of -invertibility in a Banach Jordan algebra J ; and establish the invariance of pseudospectra in a full subal- gebra of J : Furthermore, we investigate the properties of the pseudo-spectrum of an element in a Banach Jordan algebra, we examine level sets of functions and pseudo-spectral bounds. In Section 5, the study extends to linear maps preserving pseudospctrum in Banach Jordan algebras. Section 6 is about the decomposition of some elements of a Banach Jordan algebra into simpler ones in localized subalgebras. Finally, Secion 7 is dedicated to the study of Roch-Silberman theorem in a JB-algebra.