Measurable diagonalization of positive definite matrices

Abstract

In this paper we show that any positive definite matrix V with measurable entries can be written as V = U ΛU * , where the matrix Λ is diagonal, the matrix U is unitary, and the entries of U and Λ are measurable functions (U * denotes the transpose conjugate of U ).This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.MSC: 41A10; 46E35; 46G10; 47A56

Bounds on the first Betti number - an approach via Schatten norm estimates on semigroup differences

“…Remark 2.7. For completeness, we note that the afore mentioned measurable diagonalization is proved in [21] only for families (V (x)) x∈X of positive matrices. However, this easily generalizes to the case considered above.…”