Pseudospectra in Banach Jordan Algebras

The primary focus of this paper is to extend the concept of pseudospectrum from operators and matrices to elements of a unital complex Banach Jordan algebra, thereby moving from the associative to the non-associative setting. We introduce the notion of ε-invertibility in a Banach Jordan algebra J and establish the invariance of pseudospectra with respect to full subalgebras of J. We further investigate fundamental properties of the pseudospectrum of an element in a Banach Jordan algebra, including its relationship with level sets of analytic functions and pseudospectral bounds. The paper also examines linear maps that preserve pseudospectra in Banach Jordan algebras, as well as decomposition results for certain elements into simpler components within suitable localized subalgebras. Finally, we study an extension of the Roch–Silberman theorem in the setting of JB-algebras.