Inequalities Involving Unitarily Invariant Norms and Operator Monotone Functions (作用素および作用素不等式の最近の話題 研究集会報告集)
By Fumio Hiai, Xingzhi Zhan. 2002
46 citations recorded by the source corpus.
Let ∥·∥ be a unitarily invariant norm on matrices. For matrices A,B,X with A,B positive semidefinite and X arbitrary, we prove that the function t↦∥|AtXB1−t|r∥·∥|A1−tXBt|r∥ is convex on [0,1] for each r>0. This convexity result interpolates the matrix Cauchy–Schwarz inequality ∥|A1/2XB1/2|r∥2⩽∥|AX|r∥·∥|XB|r∥ due to R. Bhatia and C. Davis [Linear Algebra Appl. 223/224 (1995) 119], and also it generalizes A.W. Marshall and I. Olkin's [Pacific J. Math. 15 (1965) 241] result that the condition number ∥As∥·∥A−s∥ is increasing in s>0. We prove that if f(t) is a nonnegative operator monotone function on [0,∞) and ∥·∥ is a normalized unitarily invariant norm, then f(∥X∥)⩽∥f(|X|)∥ for every matrix X. The special case when f(t)=tr (0<r⩽1) is used to consider the monotonicity of p↦∥Ap+Bp∥1/p as well as p↦∥(Ap+Bp)1/p∥. Furthermore, we obtain some norm inequalities of Holder and Minkowski types related to the expression ∥|A|p+|B|p∥1/p. For example, comparisons are made between ∥C*A+D*B∥ and ∥|A|p+|B|p∥1/p·∥|C|q+|D|q∥1/q, where p−1+q−1=1.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/b4f457048df51cd4b7e8f70f6f0f39409a345f4b
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Norm inequalities related to the matrix geometric mean (2012, background): “For m = 2 Hiai and Zhan have shown this using the following result of Ando and Zhan [5].”
- Weak majorization inequalities and convex functions (2003, unknown): “13 when f (t) = t r , r 1 is Theorem 4 in [6].”