Operator Characterization via Projectors and Nilpotents

Abstract: This paper explores operators with countable, continuous, and hybrid spectra, focusing on both finite dimensional and infinite dimensional cases, particularly in non-Hermitian systems. For finite dimensional operators, a novel concept of analogous matrices is introduced. Here, matrices are considered analogous if they share the same projector and nilpotent structures, indicating structural equivalences beyond simple spectral similarities. A graph-based model represents these projector and nilpotent structures, offering insights for classifying analogous matrices. Additionally, the paper calculates the distinct families of analogous matrices by matrix size, establishing a tool for matrix classification. The study extends the spectral mapping theorem to multivariate functions of both Hermitian and non-Hermitian matrices, expanding the applicability of spectral theory. This theorem assumes holomorphic functions, enabling its use with a broader class of operators. The finite dimensional framework is further generalized to infinite dimensional cases, covering operators with countable spectra to deepen understanding of operator behavior. For continuous spectrum operators, this work generalizes von Neumann's spectral theorem to encompass a wider class of spectral operators, including both self-adjoint and non-self-adjoint cases. This unified approach supports a generalized spectral decomposition, facilitating application of the spectral mapping theorem across various contexts. The concept of analogous operators is also extended to continuous spectrum operators, forming a basis for their classification. Finally, operators with hybrid spectra comprising both discrete and continuous elements are examined, with analogous properties and spectral mapping explored within this context.