Topological classification of sesquilinear forms: reduction to the nonsingular case

TL;DR

This paper classifies sesquilinear forms up to topological equivalence by reducing the problem to the regularizing decomposition of associated matrices, extending known matrix regularization results.

Contribution

It establishes a complete topological classification of sesquilinear forms via their regularizing decompositions, generalizing Horn and Sergeichuk's matrix regularization to topological equivalence.

Findings

Topological equivalence characterized by regularizing decompositions

Extension of matrix regularization to sesquilinear forms

Results applicable to real and complex bilinear forms

Abstract

Two sesquilinear forms $\Phi:\mathbb C^m\times\mathbb C^m\to \mathbb C$ and $\Psi:\mathbb C^n\times\mathbb C^n\to \mathbb C$ are called topologically equivalent if there exists a homeomorphism $\varphi :\mathbb C^m\to \mathbb C^n$ (i.e., a continuous bijection whose inverse is also a continuous bijection) such that $\Phi(x,y)=\Psi(\varphi (x),\varphi (y))$ for all $x,y\in \mathbb C^m$. R.A.Horn and V.V.Sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix $A$; that is, a direct sum $SAS^*=R\oplus J_{n_1}\oplus\dots\oplus J_{n_p}$, in which $S$ and $R$ are nonsingular and each $J_{n_i}$ is the $n_i$-by-$n_i$ singular Jordan block. In this paper, we prove that $\Phi$ and $\Psi$ are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands $J_{n_i}$ and replacement of $R\in\mathbb C^{r\times r}$ by a nonsingular matrix $R'\in\mathbb C^{r\times r}$ such that $R$ and $R'$ are the matrices of topologically equivalent forms. Analogous results for real and complex bilinear forms are also obtained.