Numerical Ranges and Pseudospectra in Banach Jordan Algebras

Introduction. The numerical range is a central tool in spectral theory,
providing geometric insight into stability, perturbations, and spectral
localization in associative Banach algebras and operator theory. In the
non-associative setting of Banach Jordan algebras, however, a complete
analogue has remained underdeveloped. This work addresses this gap by
presenting a systematic theory of numerical ranges adapted to Banach
Jordan algebras, with particular emphasis on their interaction with Jordan
spectra and pseudospectra.
Methods. We introduce the quadratic numerical range WJ (a), defined
via the quadratic operator Ua, which naturally replaces left–right
multiplication in the associative case. The analysis combines Jordan functional
calculus, duality arguments, and convexity methods to study structural
properties of WJ (a) in general unital Banach Jordan algebras, with
special attention to the case of special Jordan algebras embedded in associative
C∗-algebras.
Results. We prove that the quadratic numerical range is non-empty,
bounded, closed, and convex. A central result establishes spectral containment,
showing that the Jordan spectrum is contained in the quadratic
numerical range. In addition, we obtain pseudospectral inclusion results,
demonstrating that Jordan pseudospectra are contained in explicit neighborhoods
of WJ (a). In the special Jordan setting, we show that WJ (a) is
contained in the square of the classical numerical range, thereby extending
known results from the associative theory. Several illustrative examples
in finite-dimensional settings are provided.
Conclusions. This work places numerical range theory in Banach
Jordan algebras on a rigorous and complete foundation, opening new avenues
for the study of spectral stability, perturbation theory, and nonassociative
spectral analysis.