Antidiagonal Operators, Antidiagonalization, Hollow Quasidiagonalization -- Unitary, Orthogonal, Permutation, and Otherwise - and Symmetric Spectra

Abstract: After summarizing characteristics of antidiagonal operators, we derive three direct sum decompositions characterizing antidiagonalizable linear operators - the first up to permutation-similarity, the second up to similarity, and the third up to unitary similarity. Each corresponds to a unique quasidiagonalization. We prove the permutation-similarity direct sum decomposition defines a hollow quasidiagonalization of a traceless antidiagonalizable operator and gives the real Schur decomposition of a real antisymmetric antidiagonal operator. We use this to derive an orthogonal antidiagonalization of a general real antisymmetric operator. We prove the similarity direct sum decomposition defines the eigendecomposition of an antidiagonalizable operator that is diagonalizable, and we give a characterization of this eigendecomposition. We show it also defines the Jordan canonical form for a general antidiagonalizable operator. This leads to a further characterization of antidiagonalizable operators in terms of spectral properties and a characterization as the direct sum of traceless $2 \times 2$ matrices with the exception of a single $1 \times 1$ matrix as an additional summand for operators of odd size. We discuss numerous implications of this for properties of the square of an antidiagonalizable operator, a characterization of operators that are both diagonalizable and antidiagonalizable, nilpotency of antidiagonalizable operators, unitary diagonalizations of normal antidiagonal operators, symmetric and antisymmetric antidiagonalizations, and centrosymmetric diagonalizations and antidiagonalizations. Finally, we prove the unitary similarity direct sum decomposition defines the Schur decomposition, as well as a unitary quasidiagonalization, of a unitarily antidiagonalizable operator.