Let A = (A 1 , . . . , A m ), where A 1 , . . . , A m are n × n real matrices. The real joint (p, q)-matricial range of A, Λ R p,q (A), is the set of m-tuple of q × q real matrices (B 1 , . . . , B m ) such that (X * A 1 X, . . . , X * A m X) = (I p ⊗ B 1 , . . . , I p ⊗ B m ) for some real n × pq matrix X satisfying X * X = I pq . It is shown that if n is sufficiently large, then the set Λ R p,q (A) is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H, and used to show that the joint essential (p, q)-matricial range of A is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.
Let Φ : Mn → Mn be a unital trace preserving completely positive map and A ∈ Mn be a positive definite matrix. Weak log-majorization and weak majorization between Φ(A) and A are studied. Determinantal inequalities between Φ(A) and A are obtained as a consequence. By considering special classes of unital trace preserving completely positive map, some known matrix inequalities such as Fischer's inequality are rediscovered. An affirmative answer to a question of Tam and Zhang in 2019 is given.
Let Φ : Mn → Mn be a unital trace preserving completely positive map and A ∈ Mn be a positive definite matrix. Weak log-majorization and weak majorization between Φ(A) and A are studied. Determinantal inequalities between Φ(A) and A are obtained as a consequence. By considering special classes of unital trace preserving completely positive map, some known matrix inequalities such as Fischer’s inequality are rediscovered. An affirmative answer to a question of Tam and Zhang in 2019 is given.
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