Joint spectrum of matrix and operator tuples on spaces over bi complex numbers

TL;DR

This paper extends the concept of joint spectrum and eigenvalues to matrix and operator tuples on bi-complex Hilbert spaces, revealing unboundedness properties unlike classical spectra.

Contribution

It introduces a generalized notion of joint spectrum for matrices and operators on bi-complex spaces, highlighting differences from traditional spectral properties.

Findings

Joint eigenvalues can be unbounded in bi-complex spaces.

The joint spectrum of operator tuples is not necessarily bounded.

Classical spectral boundedness does not hold in bi-complex settings.

Abstract

In this paper, we generalize the notion of joint eigenvalues and joint spectrum of matrices and operator tupples on a bi complex Hilbert space. We observe that unlike the spectrum of a bounded operator on a bi complex Hilbert space is bounded. But this is not the case here. Even the set of joint eigenvalues of matrix tuples is unbounded.