Accretive partial transpose matrices and their connections to matrix means

Accretive partial transpose (APT) matrices have been recently defined, as a natural extension of positive partial transpose (PPT) matrices. In this paper, we discuss further properties of APT matrices in a way that extends some of those properties known for PPT matrices. Among many results, we show that if A , B , X are n × n complex matrices such that A , B are sectorial with sector angle α for some α ∈ [0, π /2), and if f : (0, ∞ ) → (0, ∞ ) is a certain operator monotone function such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close="]" open="["> <m:mrow> <m:mtable class="matrix"> <m:mtr> <m:mtd columnalign="center"> <m:msup> <m:mrow> <m:mi>cos</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign="center"> <m:mi>X</m:mi> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="center"> <m:msup> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:mtd> <m:mtd columnalign="center"> <m:msup> <m:mrow> <m:mi>cos</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>α</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $[{matrix} {cos}²(α )f(A) & X \\ {X}^{{}} & {cos}²(α )f(B) {matrix}]$ is APT, Then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close="]" open="["> <m:mrow> <m:mtable class="matrix"> <m:mtr> <m:mtd columnalign="center"> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>A</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:msub> <m:mrow> <m:mi>∇</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign="center"> <m:mi>X</m:mi> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="center"> <m:msup> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> </m:mtd> <m:mtd columnalign="center"> <m:mi>f</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>A</m:mi> <m:msub> <m:mrow> <m:mi>∇</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mi>B</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> [matrix f(A)∇ ₜf(B) & X \\ X^ & f(A∇ ₜB) matrix] <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"