A matrix subadditivity inequality for f( A + B) and f( A) + f( B)

• Published in: Linear algebra and its applications Vol. 423; no. 2; pp. 512 - 518
• Main Authors: Bourin, Jean-Christophe, Uchiyama, Mitsuru
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01-06-2007; Elsevier Science
• Subjects: Algebra; Exact sciences and technology; Hermitian operators; Linear and multilinear algebra, matrix theory; Mathematics; Operator inequalities; Sciences and techniques of general use; Symmetric norms; Hermitian operators; Operator inequalities; 15A60; 47A30; Symmetric norms; t Norm; 15A60; 47A30; Hermitian matrix; Subadditivity; Matrix inequality; Positive semidefinite matrix; Block matrix; Hennitian operators; Symmetric norms; Operator inequalities
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2007.02.019

In 1999 Ando and Zhan proved a subadditivity inequality for
operator concave functions. We extend it to
all concave functions: Given positive semidefinite matrices
A,
B and a non-negative concave function
f on [0,∞),
‖
f
(
A
+
B
)
‖
⩽
‖
f
(
A
)
+
f
(
B
)
‖
for all symmetric norms (in particular for all Schatten
p-norms). The case
f
(
t
)
=
t
is connected to some block-matrix inequalities, for instance the operator norm inequality
A
X
∗
X
B
∞
⩽
max
X
|
‖
∞
;
‖
|
B
|
+
|
X
∗
|
‖
∞
}
for any partitioned Hermitian matrix.