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تسجيل مستخدم جديدPositive Functionals and Hessenberg Matrices
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Not every positive functional defined on bi-variate polynomials of a prescribed degree bound is represented by the integration against a positive measure. We isolate a couple of conditions filling this gap, either by restricting the class of polynomials to harmonic ones, or imposing the vanishing of a defect indicator. Both criteria offer constructive cubature formulas and they are obtained via well known matrix analysis techniques involving either the dilation of a contractive matrix to a unitary one or the specific structure of the Hessenberg matrix associated to the multiplier by the underlying complex variable.
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We bring in some new notions associated with $2times 2$ block positive semidefinite matrices. These notions concern the inequalities between the singular values of the off diagonal blocks and the eigenvalues of the arithmetic mean or geometric mean o
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Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices
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