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Pseudospectra and matrix behaviour

Thomas Ransford — 2010

Banach Center Publications

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.

Subharmonicity in von Neumann algebras

Thomas RansfordMichel Valley — 2005

Studia Mathematica

Let ℳ be a von Neumann algebra with unit 1 . Let τ be a faithful, normal, semifinite trace on ℳ. Given x ∈ ℳ, denote by μ t ( x ) t 0 the generalized s-numbers of x, defined by μ t ( x ) = inf||xe||: e is a projection in ℳ i with τ ( 1 - 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, λ 0 t l o g μ s ( f ( λ ) ) d s is a subharmonic function on D. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.

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