Introduction to Matrix Analysis and Applications by Fumio Hiai & Dénes Petz

Author:Fumio Hiai & Dénes Petz
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Since

it follows that maps into itself. Hence .

Theorem 4.40

Let be a non-linear matrix convex function on . Then there exists a unique probability Borel measure on such that

Proof: To prove this statement, we use the result due to Kraus that if is a matrix convex function on , then is and is matrix monotone on for every . Then we may assume that by considering . Since is a non-constant matrix monotone function on , by Theorem 4.38 there exists a probability Borel measure on such that

Since is easily seen, we have

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