On the weighted geometric mean of accretive matrices
- PDF / 1,192,023 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 10 Downloads / 180 Views
Annals of Functional Analysis https://doi.org/10.1007/s43034-020-00094-6 ORIGINAL PAPER
On the weighted geometric mean of accretive matrices Yassine Bedrani1 · Fuad Kittaneh1 · Mohammed Sababheh2 Received: 16 July 2020 / Accepted: 7 September 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, we discuss new inequalities for accretive matrices through nonstandard domains. In particular, we present several relations for Ar and A♯r B , when A, B are accretive and r ∈ (−1, 0) ∪ (1, 2). This complements the well-established discussion of such quantities for accretive matrices when r ∈ [0, 1], and provides accretive versions of known results for positive matrices. Among many other results, we show that the accretive matrices A, B satisfy
ℜ(A♯r B) ≤ ℜA♯r ℜB, r ∈ (−1, 0) ∪ (1, 2). This, and other results, gain their significance due to the fact that they are reversed when r ∈ (0, 1). Keywords Sectorial matrix · Accretive matrix · Geometric mean · Positive matrix · Positive linear map Mathematics Subject Classification 15A45 · 15A60 · 47A30 · 47A63 · 47A64
1 Introduction Let Mn be the algebra of all n × n complex matrices. For A ∈ Mn , recall the Cartesian decomposition Communicated by Jean-Christophe Bourin. * Mohammed Sababheh [email protected] Yassine Bedrani [email protected] Fuad Kittaneh [email protected] 1
Department of Mathematics, The University of Jordan, Amman, Jordan
2
Department of Basic Sciences, Princess Sumaya University for Technology, Amman 11941, Jordan
Vol.:(0123456789)
Y. Bedrani et al.
A = ℜA + iℑA, with ℜA =
A − A∗ A + A∗ and ℑA = , 2 2i
where ℜA is the real part of A and ℑA is the imaginary part of A. We say that A is positive semidefinite (written A ≥ 0 ) if ⟨Ax, x⟩ ≥ 0 for all vectors x ∈ ℂn , and that A is positive (written A > 0 ) if ⟨Ax, x⟩ > 0 for all nonzero vectors x ∈ ℂn . The class of positive matrices will be denoted by M+n . A more general class of matrices than that of positive ones is the so-called accretive matrices. A matrix A ∈ Mn is said to be accretive when ℜA > 0. It is clear that when A is positive, it is necessarily accretive. For two Hermitian matrices A, B ∈ Mn , we say that A ≤ B (or A < B ) if B − A ≥ 0 (or B − A > 0 ). The relation A ≤ B defines a partial ordering on the class of Hermitian matrices. The numerical radius w(A) and the operator norm ‖A‖ of A ∈ Mn are defined, respectively, by and
w(A) = max{�⟨Ax, x⟩� ∶ x ∈ ℂn , ‖x‖ = 1}
‖A‖ = max{�⟨Ax, y⟩� ∶ x, y ∈ ℂn , ‖x‖ = ‖y‖ = 1}.
Recall that a norm ||| ⋅ ||| on Mn is unitarily invariant if |||UAV||| = |||A||| for any A ∈ Mn and for all unitary matrices U, V ∈ Mn. For two matrices A, B ∈ M+n , the weighted geometric mean of A and B is defined as [9] ( 1 )r 1 1 1 A♯r B = A 2 A− 2 BA− 2 A 2 , where 0 ≤ r ≤ 1. (1.1) This matrix mean is one among many well-defined matrix means. Other known and easily defined matrix means are the arithmetic and harmonic means for A, B ∈ M+n , defined, respectively by
A∇r B = (1 − r)A + rB, A!r B = ((1 − r)A−1 + rB−1 )−1 , r ∈ [0, 1].
Data Loading...
