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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.

Sufficient conditions to be exceptional

Charles R. Johnson, Robert B. Reams (2016)

Special Matrices

A copositive matrix A is said to be exceptional if it is not the sum of a positive semidefinite matrix and a nonnegative matrix. We show that with certain assumptions on A−1, especially on the diagonal entries, we can guarantee that a copositive matrix A is exceptional. We also show that the only 5-by-5 exceptional matrix with a hollow nonnegative inverse is the Horn matrix (up to positive diagonal congruence and permutation similarity).

Sulle forme polarizzanti i coefficienti del polinomio caratteristico di una matrice

Renzo Mazzocco (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The multilinear forms, obtained by polarizing the coefficients of the characteristic polynomial of a matrix, are considered. A general relation (formula A) between such forms is proved. It follows in particular a rational expression for the above-mentioned coefficients (formula C), which is in a sense analogous to Newton's formulas, but with the use of the determinant function instead of the trace function.

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