Fixed points of holomorphic transformations of operator balls

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.

💡 Research Summary

The paper develops a novel geometric method for establishing fixed‑point theorems for families of holomorphic automorphisms acting on the operator ball, i.e., the set B = { T ∈ B(H) : ‖T‖ ≤ 1 } of bounded operators on a Hilbert space H equipped with the Carathéodory distance. The authors first show that (B, d_C) is a complete, non‑positively curved metric space, and that holomorphic automorphisms of B are isometries for this metric. By introducing an “internal compression” technique they prove that any uniformly bounded subgroup G of such automorphisms admits a common fixed point in the interior of B, provided the elements of G stay a uniform distance away from the boundary (i.e., there exists c < 1 with ‖g‖ ≤ c for all g ∈ G). This result extends classical fixed‑point theorems (Browder–Kirk, Kirk–Sims) to the infinite‑dimensional holomorphic setting, where the maps are not merely non‑expansive but also complex‑analytic and invertible.

The second major contribution applies the fixed‑point theorem to representation theory. Let π : Γ → GL(H) be a bounded representation of a group Γ on a real or complex Hilbert space that preserves an indefinite quadratic form Q with finitely many negative squares (a Pontryagin space structure). The group of Q‑preserving operators, O(Q) or U(Q), acts on the operator ball by conjugation, and each element of π(Γ) is a holomorphic automorphism of B. By the fixed‑point theorem, π(Γ) has a common fixed point inside B; this fixed point corresponds to a positive‑definite operator S such that Sπ(g)S⁻¹ is Q‑orthogonal (real case) or unitary (complex case) for every g ∈ Γ. Consequently, any bounded representation that admits a finite‑negative‑index invariant form is similar to an orthogonal (or unitary) representation; in other words, it is orthogonalizable or unitarizable.

The paper concludes with a discussion of further directions. The authors suggest extending the method to other operator balls (e.g., Schatten‑p classes), investigating the case where the invariant form has infinitely many negative squares, and exploring applications to nonlinear operator groups arising in quantum mechanics and signal processing. Overall, the work bridges fixed‑point theory and representation theory in infinite‑dimensional holomorphic contexts, providing a powerful tool for converting bounded representations with indefinite metrics into genuinely orthogonal or unitary ones.