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We study the extent to which the pseudospectra of a matrix determine other aspects of its behaviour, such as the growth of its powers and its unitary equivalence class.

We construct two Banach algebras, one which contains analytic semigroups ${\left(a^z\right)}_{Rez>0}$ such that $|a^{1+iy}|\to \infty$ arbitrarily slowly as $\left|y\right|\to \infty$, the other which contains ones such that $|a^{1+iy}|\to \infty$ arbitrarily fast

Let ℳ be a von Neumann algebra with unit $1_{ℳ}$. Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by ${\mu }_t{\left(x\right)}_{t\ge 0}$ the generalized s-numbers of x, defined by ${\mu }_t\left(x\right)$ = inf||xe||: e is a projection in ℳ i with $\tau (1_{ℳ}-e)$ ≤ t (t ≥ 0). We prove that, if D is a complex domain and f:D → ℳ is a holomorphic function, then, for each t ≥ 0, $\lambda \mapsto {\int }_0^{t}log{\mu }_s\left(f\left(\lambda \right)\right)ds$ is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.