From Clouatre-Ostermann-Ransford to Okubo-Ando

Abstract

Let $\mathcal A$ be a unital operator algebra and let $\theta: \mathcal A \to B(\mathcal H)$ be a continuous unital homomorphism. We prove that for all $n \in \mathbb{N}$ and all $\beta$ in the dual space of $\mathcal{A}$, \begin{equation*} \|\theta^{(n)}\| \le \max(1, \|\theta^{(n)} + \beta^{(n)} I \|). \end{equation*} Here, $\theta^{(n)}: M_n(\mathcal A) \to B(\mathcal H^n)$ denotes the $n$-th matrix ampliation of $\theta$. This extends a result of Clou\^atre, Ostermann and Ransford (the case $n=1$), who were motivated by Crouzeix's conjecture. Our result allows us to control the completely bounded norm of $\theta$, which in turn has dilation theoretic consequences.   As an application, we obtain a new proof of the similarity theorem of Okubo and Ando, which says that if $T \in B(\mathcal H)$ is an operator of class $C_\rho$, where $\rho \ge 1$, then there exists an invertible operator $S$ such that $\|S^{-1} T S \| \le 1$ and such that $\|S\| \|S^{-1}\| \le \rho$. The conclusion of the Okubo--Ando theorem implies that the inequality \[ \|f(T)\| \le \rho \|f\|_{\overline{\mathbb D}} \] holds for all polynomials $f$. A direct proof of this inequality was recently given by Clou\^atre, Ostermann and Ransford. Our main result shows that this inequality in fact holds for matrix-valued polynomials, so that the existence of the similarity $S$ follows from Paulsen's similarity theorem, which says that an operator $T$ is completely polynomially bounded with constant $\rho$ if and only if there exists an invertible operator $S$ such that $\|S^{-1} T S \| \le 1$ and such that $\|S\| \|S^{-1}\| \le \rho$.   The key to proving our main result is to adapt a variational argument of Caldwell, Greenbaum and Li to matrix-valued holomorphic functions. Rather than working with biholomorphic automorphisms of the disc, we work with Potapov--M\"obius transforms $m_A$, where $A$ belongs to the unit ball of $M_n(\mathbb C)$. A slightly simplified version of our key lemma, which is shown using this technique, then reads as follows: If $R \in M_n(B(\mathcal H))$ satisfies $\|R\| > 1$ and $\|m_{A \otimes I}(R)\| \le \|R\|$ for all $A$ close to $0$ and if $x \in \mathcal H^n$ with $\|x\| = 1$ and $\|R x\| = \|R\|$, then \[
\langle R x, (A \otimes I) x \rangle = 0 \] for all $A \in M_n(\mathcal C)$.