Inequalities for sector matrices and positive linear maps

Abstract

Ando proved that if A, B are positive definite, then for any positive linear map Φ, it holds Φ(A#λB) ≤ Φ(A)#λΦ(B), where A#λB, 0 ≤ λ ≤ 1, means the weighted geometric mean of A, B. Using the recently defined geometric mean for accretive matrices, Ando’s result is extended to sector matrices. Some norm inequalities are considered as well.
 

Further properties of accretive matrices

“…then A will be called a sectorial matrix and we simply write A ∈ Π α n , where I(z) denotes the imaginary part of z. We refer the reader to [6,7,8,9,14,25,30,31,40,44,45] for an almost comprehensive overview of the progress that has been made in studying inequalities in Π n . We emphasize here that whenever we use the notation Π α n in this paper, we implicitly understand that 0 ≤ α < π 2 .…”

Further Properties of Accretive Matrices

“…then A will be called a sectorial matrix and we simply write A ∈ Π α n , where I(z) denotes the imaginary part of z. We refer the reader to [5,6,7,8,13,22,26,28,36,40,42] for an almost comprehensive overview of the progress that has been made in studying inequalities in Π n . We emphasize here that whenever we use the notation Π α n in this paper, we implicitly understand that 0 ≤ α < π 2 .…”

Inequalities for sector matrices with negative power

“…Moreover, W(A) ⊆ S α implies W(X * AX) ⊆ S α for any nonzero n × m matrix X, thus W(A −1 ) ⊆ S α . Recently, Tan and Chen [21] also proved that for any positive linear map Φ, W(A) ⊆ S α implies that W(Φ(A)) ⊆ S α . Recent developments on sector matrices can be found in [9,11,[13][14][15][16][17]21].…”

“…Recently, Tan and Chen [21] also proved that for any positive linear map Φ, W(A) ⊆ S α implies that W(Φ(A)) ⊆ S α . Recent developments on sector matrices can be found in [9,11,[13][14][15][16][17]21].…”