Browsing by Subject "Euclidean Jordan algebra"
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Item A Hölder type inequality and an interpolation theorem in Euclidean Jordan algebras(2018-09-14) Gowda, Muddappa SeetharamaIn a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element x we associate the eigenvalue vector λ(x) whose components are the eigenvalues of x written in the decreasing order. For any p ∈ [1,∞], we define the spectral p-norm of x to be the p-norm of λ(x) in Rⁿ. In this paper, we show that ||x ◦ y||1 ≤ ||x|| ||y||q, where x ◦ y denotes the Jordan product of two elements x and y in V and q is the conjugate of p. For a linear transformation on V, we state and prove an interpolation theorem relative to these spectral norms. In addition, we compute/estimate the norms of Lyapunov transformations, quadratic representations, and positive transformations on V.Item On the connectedness of spectral sets and irreducibility of spectral cones in Euclidean Jordan algebras(2018) Gowda, Muddappa; Jeong, JuyoungLet V be a Euclidean Jordan algebra of rank n. A set E in V is said to be a spectral set if there exists a permutation invariant set Q in Rn such that E = λ−1(Q), where λ : V → Rn is the eigenvalue map that takes x ∈ V to λ(x) (the vector of eigenvalues of x written in the decreasing order). If the above Q is also a convex cone, we say that E is a spectral cone. This paper deals with connectedness and arcwise connectedness properties of spectral sets. By relying on the result that in a simple Euclidean Jordan algebra, every eigenvalue orbit [x] := {y : λ(y) = λ(x)} is arcwise connected, we show that if a permutation invariant set Q is connected (arcwise connected), then λ−1(Q) is connected (respectively, arcwise connected). A related result is that in a simple Euclidean Jordan algebra, every pointed spectral cone is irreducible.Item A Riesz-Thorin type interpolation theorem in Euclidean Jordan algebras(2019-05-07) Gowda, M. Seetharama; Sznajder, RomanIn a Euclidean Jordan algebra V of rank n which carries the trace inner product, to each element a we associate the eigenvalue vector λ(a) in Rn whose components are the eigenvalues of a written in the decreasing order. For any p∈[1,∞], we define the spectral p-norm of a to be the p-norm of λ(a) in Rⁿ. In a recent paper, based on the K-method of real interpolation theory and a majorization technique, we described an interpolation theorem for a linear transformation on V relative to the same spectral norm. In this paper, using standard complex function theory methods, we describe a Riesz-Thorin type interpolation theorem relative to two different spectral norms. We illustrate the result by estimating the norms of certain special linear transformations such as Lyapunov transformations, quadratic representations, and positive transformations.Item Spectral sets and functions on Euclidean Jordan algebras(2017-01-01) Jeong, Juyoung; Gowda, Muddappa S; Mathematics and Statistics; Mathematics, AppliedThis thesis studies spectral and weakly spectral sets/functions on Euclidean Jordan algebras. These are generalizations of similar well-known concepts on the algebras of real symmetric and complex Hermitian matrices. A spectral set in a Euclidean Jordan algebra V is the inverse image of a permutation invariant set in R^n under the eigenvalue map (which takes an element x in V to its eigenvalue vector in R^n consisting of eigenvalues of x written in the decreasing order). A spectral function on V is the composition of a permutation invariant function on R^n and the eigenvalue map. In this thesis, we study properties of such sets/functions and show how they are related to algebra automorphisms and majorization. We show they are indeed invariant under algebra automorphisms of V, hence weakly spectral with converse holding when V is essentially simple. For a spectral set K, we discuss the transfer principle and a related metaformula. When K is also a cone, we show that the dual of K is a spectral cone under certain conditions. We also discuss the dimension of K, and characterize the pointedness/solidness of K. Specializing, we study permutation invariant (proper) polyhedral cones in R^n. We show that the Lyapunov rank of such a cone divides n. Lastly, we study Schur-convexity of a spectral function and describe some applications.