Weighted geometric mean of two accretive matrices
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Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00105-6 ORIGINAL PAPER
Weighted geometric mean of two accretive matrices Junjian Yang1,2,3,4 · Linzhang Lu1,5 Received: 29 October 2020 / Accepted: 10 November 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this note, we prove the equalities for the weighted geometric mean of two accretive matrices A and B: 1
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A♯𝜈 B = B♯1−𝜈 A = A 2 (A− 2 BA− 2 )𝜈 A 2 = B 2 (B− 2 AB− 2 )1−𝜈 B 2 ,
0 < Re 𝜈 < 1,
which inherit the same expressions as positive semidefinite matrices. We also prove some weighted AM–GM–HM inequalities for such accretive matrices. Keywords Weighted arithmetic mean · Weighted geometric mean · Weighted harmonic mean · Accretive matrices Mathematics Subject Classification 47A63 · 15A45
Communicated by Jean-Christophe Bourin. * Junjian Yang [email protected]
Linzhang Lu [email protected]; [email protected]
1
School of Mathematical Sciences, Guizhou Normal University, Guiyang, People’s Republic of China
2
School of Mathematics and Statistics, Hainan Normal University, Haikou, People’s Republic of China
3
Key Laboratory Of Data Science And Intelligence Education (Hainan Normal University), Ministry of Education, Haikou, People’s Republic of China
4
Key Laboratory of Computational Science and Application of Hainan Province, Haikou, People’s Republic of China
5
School of Mathematical Sciences, Xiamen University, Xiamen, People’s Republic of China
Vol.:(0123456789)
J. Yang, L. Lu
1 Introduction Let 𝕄n denote the set of n × n complex matrices. Then every A ∈ 𝕄n can be written in the form ∗ ∗ and Im A = A−A . A = Re A + i Im A, with Re A = A+A 2 2i This is the so-called Cartesian decomposition of A (see, e.g., [1, p. 6] and [6, p. 7]), where the matrices Re A and Im A are the real and imaginary parts of A. If A ∈ 𝕄n is positive semidefinite (respectively, definite), then we write A ≥ 0 (A > 0) . For two Hermitian matrices A, B of the same size, A ≥ B (respectively, A > B ) means that A − B ≥ 0 (respectively, A − B > 0). When 𝛾 is real, the principal power function z ↦ z𝛾 is the map from ℂ �(−∞, 0] to ℂ defined by z𝛾 = r𝛾 ei𝛾𝜃 where z = rei𝜃 , r > 0 and −𝜋 < 𝜃 < 𝜋. The definition can be extended to all complex matrices, that is, for any complex matrix A ∈ 𝕄n with all its eigenvalues in ℂ �(−∞, 0] , the principal power of A is defined by
A𝛾 =
1 (zI − A)−1 z𝛾 dz, 2𝜋i ∫Γ
where Γ is a closed contour enclosing each eigenvalue of A and avoiding (−∞, 0] . For fundamentals of matrix functions, see [4][5]. For 𝛼 ∈ [0, 𝜋2 ) , let S𝛼 be the sector in the complex plane given by
S𝛼 = {z ∈ ℂ|Re z > 0, |Im z| ≤ (Re z) tan 𝛼}.
The numerical range of an n × n matrix A is defined by
W(A) = {x∗ Ax|x ∈ ℂn , x∗ x = 1}. The class of matrices A with W(A) ⊂ S𝛼 is called sector matrices. Sector matrices have been the subject of a number of recent papers [2, 7, 8, 9, 10–12, 16]. It is clear that W(A) ⊂ S𝛼 implies W(A−1 ) ⊂ S𝛼 (see, e.g., [11]). If A, B ∈ 𝕄n satisfy W(A), W(B) ⊂ S𝛼 , then W(A + B) ⊂ S𝛼 (see,
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