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Studying the various properties of MIN and MAX matrices - elementary vs. more advanced methods

Mika Mattila, Pentti Haukkanen (2016)

Special Matrices

Let T = {z1, z2, . . . , zn} be a finite multiset of real numbers, where z1 ≤ z2 ≤ · · · ≤ zn. The purpose of this article is to study the different properties of MIN and MAX matrices of the set T with min(zi , zj) and max(zi , zj) as their ij entries, respectively.We are going to do this by interpreting these matrices as so-called meet and join matrices and by applying some known results for meet and join matrices. Once the theorems are found with the aid of advanced methods, we also consider whether...

Subharmonicity in von Neumann algebras

Thomas Ransford, Michel 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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