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Journal of Lie Theory 30 (2020), No. 2, 473--488
Copyright Heldermann Verlag 2020

Hadamard Semigroups of Off-Diagonal Constant Matrices

Yongdo Lim
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea

The convex cone of positive semidefinite matrices of fixed size forms a commutative topological semigroup under the Hadamard product. In this paper we consider the closed subsemigroup of off-diagonal constant matrices, matrices having the same value in the off-diagonal positions, and its compact and convex subsemigroup of matrices with diagonal entries in the unit interval. Several results on these topological semigroups are presented: the group of units, (Loewner) ordered semigroup structures, one-parameter semigroups. An application of Hadamard powers obtained by FitzGerald and Horn and related open problems on Euclidean Jordan algebras are discussed.

Keywords: Positive semidefinite matrix, Schur product theorem, Hadamard semigroup, off-diagonal constant matrix, topological semigroup, Loewner order, one-parameter semigroup, infinitely divisible matrix, Euclidean Jordan algebra, spin factor.

MSC: 22A20, 22A15, 15B48, 47L07.

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