Function theory of antilinear operators

TL;DR

This paper develops a function theory for antilinear operators, exploring their spectral properties, associated function spaces, and a novel orthogonalization process linked to biradial measures.

Contribution

It introduces a new function space for antilinear operators, establishes spectral mapping theorems, and connects Jacobi parameters with biradial measures through orthogonalization.

Findings

Spectral mapping theorems for antilinear operators are established.

Functions associated with these operators have a biradial character.

A correspondence between Jacobi parameters and biradial measures is demonstrated.

Abstract

Unlike in complex linear operator theory, polynomials or, more generally, Laurent series in antilinear operators cannot be modelled with complex analysis. There exists a corresponding function space, though, surfacing in spectral mapping theorems. These spectral mapping theorems are inclusive in general. Equality can be established in the self-adjoint case. The arising functions are shown to possess a biradial character. It is shown that to any given set of Jacobi parameters corresponds a biradial measure yielding these parameters in an iterative orthogonalization process in this function space, once equipped with the corresponding $L^2$ structure.