Basic Algebra of Antilinear Operators and Some Applications. I

Abstract

The primary aim of this work is to find the canonical (i.e., simplest) form of antilinear operators which is the analog of the diagonal form of linear ones, as well as to obtain that class of antilinear operators which corresponds to the class of normal, i.e., diagonalizable, linear ones. To achieve this aim two basic tools are used: polar factorization of an arbitrary antilinear operator into a linear, Hermitian, positive, semidefinite, and anti-unitary operator Âa = Ĥ1Ûa = ÛaĤ2, and representation of antilinear operators by antilinear matrices, which are products of a matrix factor transforming by unitary congruence transformations and the operation of conjugation which is the same for all antilinear operators and all bases. The canonical form is defined as the simplest form of the matrix factor. The criteria of simplicity are: a quasi-diagonal form with smallest possible submatrices, a maximal number of zeros in them, and as many positive numbers as possible among the nonzero elements. It is found that the analog of the diagonal form of linear operators is the second-order canonical form consisting of a diagonal part with nonnegative elements, which is as large as possible, and of two-by-two submatrices with zeros on the diagonal and with at least one positive element. The operators having this form are those whose polar factors can be simultaneously canonical, taking for the anti-unitary factor, essentially the Wigner canonical form. These antilinear operators are called normal ones, and they can also be defined by the following relations between the polar factors: [Ĥ1, Ĥ2]− = 0 and [Ĥ1, Ûa2]−=0 or by the single commutator, [Âa, (Âa†)2]−=0. A simple procedure to obtain the canonical form of a given normal antilinear operator is developed. A few applications of the results obtained are outlined. They belong to different fields such as electric network theory, quantum mechanics, and self-consistent Hartree-Bogoliubov theory.