Submultiplicativity vs subadditivity for unitarily invariant norms

Fumio Hiai; Xingzhi Zhan

doi:10.1016/j.laa.2003.08.006

Submultiplicativity vs subadditivity for unitarily invariant norms

Fumio Hiai, Xingzhi Zhan

School of Mathematical Sciences

Tohoku University

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Let A, B be nonzero positive semidefinite matrices. We prove that ∥AB∥/∥A∥ ∥B∥ ≤ ∥A + B∥/∥A∥+∥B∥,∥A ο B∥/∥A∥ ∥B∥ ≤ ∥A + B∥/∥A∥ + ∥B∥ for any unitarily invariant norm with ∥diag(1, 0,..., 0)∥ ≥ 1. Some related inequalities are derived.

Original language	English
Pages (from-to)	155-164
Number of pages	10
Journal	Linear Algebra and Its Applications
Volume	377
Issue number	1-3
DOIs	https://doi.org/10.1016/j.laa.2003.08.006
State	Published - 15 Jan 2004

Keywords

Hadamard product
Majorization
Matrix Young inequality
Positive semidefinite matrix
Subadditivity
Submultiplicativity
Unitarily invariant norm

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10.1016/j.laa.2003.08.006

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title = "Submultiplicativity vs subadditivity for unitarily invariant norms",

abstract = "Let A, B be nonzero positive semidefinite matrices. We prove that ∥AB∥/∥A∥ ∥B∥ ≤ ∥A + B∥/∥A∥+∥B∥,∥A ο B∥/∥A∥ ∥B∥ ≤ ∥A + B∥/∥A∥ + ∥B∥ for any unitarily invariant norm with ∥diag(1, 0,..., 0)∥ ≥ 1. Some related inequalities are derived.",

keywords = "Hadamard product, Majorization, Matrix Young inequality, Positive semidefinite matrix, Subadditivity, Submultiplicativity, Unitarily invariant norm",

author = "Fumio Hiai and Xingzhi Zhan",

year = "2004",

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doi = "10.1016/j.laa.2003.08.006",

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T1 - Submultiplicativity vs subadditivity for unitarily invariant norms

AU - Hiai, Fumio

AU - Zhan, Xingzhi

PY - 2004/1/15

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N2 - Let A, B be nonzero positive semidefinite matrices. We prove that ∥AB∥/∥A∥ ∥B∥ ≤ ∥A + B∥/∥A∥+∥B∥,∥A ο B∥/∥A∥ ∥B∥ ≤ ∥A + B∥/∥A∥ + ∥B∥ for any unitarily invariant norm with ∥diag(1, 0,..., 0)∥ ≥ 1. Some related inequalities are derived.

AB - Let A, B be nonzero positive semidefinite matrices. We prove that ∥AB∥/∥A∥ ∥B∥ ≤ ∥A + B∥/∥A∥+∥B∥,∥A ο B∥/∥A∥ ∥B∥ ≤ ∥A + B∥/∥A∥ + ∥B∥ for any unitarily invariant norm with ∥diag(1, 0,..., 0)∥ ≥ 1. Some related inequalities are derived.

KW - Hadamard product

KW - Majorization

KW - Matrix Young inequality

KW - Positive semidefinite matrix

KW - Subadditivity

KW - Submultiplicativity

KW - Unitarily invariant norm

UR - https://www.scopus.com/pages/publications/0242303536

U2 - 10.1016/j.laa.2003.08.006

DO - 10.1016/j.laa.2003.08.006

M3 - 文章

AN - SCOPUS:0242303536

SN - 0024-3795

VL - 377

SP - 155

EP - 164

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

ER -

Submultiplicativity vs subadditivity for unitarily invariant norms

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Keywords

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