Fixed points of holomorphic transformations of operator balls

TL;DR

This paper introduces a novel technique for establishing fixed point theorems for holomorphic transformations on operator balls, with applications to the orthogonalizability and unitarizability of bounded Hilbert space representations.

Contribution

It develops a new method for fixed point proofs in operator theory and applies it to characterize when certain Hilbert space representations are orthogonalizable or unitarizable.

Findings

Established fixed point theorems for holomorphic transformations of operator balls.

Proved that bounded representations with invariant indefinite quadratic forms are orthogonalizable or unitarizable.

Provided criteria for the similarity of representations to orthogonal or unitary forms.

Abstract

A new technique for proving fixed point theorems for families of holomorphic transformations of operator balls is developed. One of these theorems is used to show that a bounded representation in a real or complex Hilbert space is orthogonalizable or unitarizable (that is similar to an orthogonal or unitary representation), respectively, provided the representation has an invariant indefinite quadratic form with finitely many negative squares.