Title of article :
THE n-TH RESIDUAL RELATIVE OPERATOR ENTROPY R[n] x;y(AjB) and THE n-TH OPERATOR VALUED DIVERGENCE
Author/Authors :
Tohyama, Hiroaki Department of Life Science and Informatics - Maebashi Institute of Technology - Kamisadori - Maebashi - Gunma, Japan , Kamei, Eizaburo Sakuragaoka - Kanmakicho - Kitakaturagi-gun - Nara, Japan , Watanabe, Masayuki Maebashi Institute of Technology - Kamisadori - Maebashi - Gunma, Japan
Abstract :
We introduced the n-th residual relative operator entropy R[n]
x;y(AjB) and showed its mono-
tone property, for example, R[n]
x;x(AjB) R[n]
x;y(AjB) R[n]
y;y(AjB) and R[n]
x;x(AjB) R[n]
y;x(AjB)
R[n]
y;y(AjB) for x y if A B or n is odd. The n-th residual relative operator entropy R[n]
x;y(AjB) is
not symmetric on x and y, that is, R[n]
x;y(AjB) ̸= R[n]
y;x(AjB) for n 2 while R[1]
x;y(AjB) = R[1]
y;x(AjB).
In this paper we compare R[n]
x;y(AjB) with R[n]
y;x(AjB) and give the relations between R[n]
x;y(AjB) and the
n-th operator divergence Δ[n]
i;x(AjB). In this process, we nd another operator divergence Δ
[n]
i;x(AjB)
which is similar to Δ[n]
i;x(AjB) but not the same.
Keywords :
the n-th operator valued divergence , the n-th residual relative operator entropy , The n-th relative operator entropy
Journal title :
journal of the iranian mathematical society
