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Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls. (English) Zbl 1185.47061

In this article, the authors provide conditions which imply that uniformly bounded representations of a group on a Hilbert space are unitarizable. The authors first show that, if the open unit ball \(\mathcal{B}\) of the space of bounded linear operators from a finite-dimensional Hilbert space \(K\) to a separable Hilbert space \(H\) is endowed with a hyperbolic metric \(\rho\), then every weakly compact, \(\rho\)-convex subset of \(\mathcal{B}\) with positive diameter contains a non-diametral point. As a consequence of this, the authors prove that, if \(K\) is a finite-dimensional Hilbert space, \(H\) a separable Hilbert space, and \(G\) a group of biholomorphic automorphisms of \(\mathcal{B}\), then \(G\) has a common fixed point in \(\mathcal{B}\) if and only if at least one orbit of \(G\) is separated from the boundary of \(\mathcal{B}\) (i.e., if and only if there exists \(A\in \mathcal{B}\) such that the set \(\{g(A):g\in G\} \subset r\mathcal{B}\) for some \(0\leq r< 1\)). This fixed point result is then used to prove that a bounded representation \(\pi\) of a group \(G\) on a Hilbert space \(H\) is similar to a unitary representation if \(\pi\) preserves a quadratic form \(\eta (x) = \| Px\|^2-\| Qx\|^2\), where \(x\in H\) and \(P\) and \(Q\) are orthogonal projections on \(H\) with \(P+Q=I\) and \(\dim(QH)<\infty\). The authors apply this to find dual pairs of invariant subspaces in Pontryagin spaces.

MSC:

47H10 Fixed-point theorems
46T25 Holomorphic maps in nonlinear functional analysis
46G20 Infinite-dimensional holomorphy
46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)
51M10 Hyperbolic and elliptic geometries (general) and generalizations
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