Some generalizations for singular value inequalities of compact operators
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Tusi Mathematical Research Group
ORIGINAL PAPER
Some generalizations for singular value inequalities of compact operators Wasim Audeh1 Received: 15 June 2020 / Accepted: 5 October 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract Audeh and Kittaneh have proved the following. Let X, Y and Z be compact operX Z 0. Then ators on a complex separable Hilbert space such that Z Y sj ðZÞ sj ðX YÞ for j ¼ 1; 2; . . . In this paper, we provide a considerable generalization of this singular value inequality, which states that: Let X, Y and Z be compact operators on a X Z 0 and let A, B be bounded complex separable Hilbert space such that Z Y linear operators on a complex separable Hilbert space. Then n o sj ðAZB Þ max k Ak2 ; kBk2 sj ðX YÞ for j ¼ 1; 2; . . . Several generalizations for singular value inequalities of compact operators are also given. Keywords Singular value Compact operator Inequality Positive operator
Mathematics Subject Classification 15A18 15A42 47A63 47B07 47B15
Communicated by Yiu-Tung Poon. & Wasim Audeh [email protected] 1
University of Petra, Amman, Jordan
W. Audeh
1 Introduction Denote the algebra of all bounded linear operators on a complex Hilbert space H by BðHÞ, and the algebra of all compact operators on H by KðHÞ. The singular values of A 2 KðHÞ are the eigenvalues of its absolute value j Aj ¼ ðA AÞ1=2 , denoted by s1 ðAÞ s2 ðAÞ and are ordered according to multiplicity. We define k Ak ¼ s1 ðAÞ, where k:k denotes the usual operator norm on BðHÞ. Singular values of A; B 2 KðHÞ have the following properties: 1. sj ðUAVÞ ¼ sj ðAÞ ¼ sj ðA Þ ¼ sj ðj AjÞ ¼ sj ðjA jÞ, for j ¼ 1; 2; . . . where U and V are unitary. 2. sj ðAA Þ ¼ sj ðA AÞ for j ¼ 1; 2; . . . A 0 0 B A 0 3. sj ¼ sj for j ¼ 1; 2; . . ., where the singular values of 0 B A 0 0 B A 0 consist of those of A together with those of B, where is defined on 0 B H H. 4. By applying Weyl’s monotonicity principle (see, e.g., [4, p. 63] or [7, p. 26]), if A; B 2 KðHÞ such that A; B 0 and A B, then sj ðAÞ sj ðBÞ for j ¼ 1; 2; . . . The classical arithmetic-geometric mean inequality for numbers states that if a, b be pffiffiffiffiffi positive numbers, then ab aþb 2 , we conclude that for complex numbers a, b we 2
2
have jabj jaj þ2 jbj . Bhatia and Kittaneh in [5] extend this inequality for compact operators as follows: If A; B 2 KðHÞ, then 2sj ðAB Þ sj ðA A þ B BÞ
ð1Þ
for j ¼ 1; 2; . . . Tao has proved in [10] an form of inequality (1), which equivalent A B 0, then states that if A; B; C 2 KðHÞ such that B C A B 2sj ðBÞ sj ð2Þ B C for j ¼ 1; 2; . . . Moreover, Bhatia and Kittaneh have proved in [6] related inequalities: 1. If A; B 2 KðHÞ such that A is self-adjoint, B 0, and A B, then sj ðAÞ sj ðB BÞ
ð3Þ
sj ðAB þ BA Þ sj ððAA þ BB Þ ðAA þ BB ÞÞ
ð4Þ
for j ¼ 1; 2; . . . 2. If A; B 2 KðHÞ, then
for j ¼ 1; 2; . . .
Some generalizations for singular value inequalities...
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