@article { author = {Tohyama, H. and Kamei, E. and Watanabe, M.}, title = {The $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and the $n$-th operator valued divergence}, journal = {Journal of the Iranian Mathematical Society}, volume = {2}, number = {2}, pages = {71-79}, year = {2021}, publisher = {Iranian Mathematical Society}, issn = {2717-1612}, eissn = {2717-1612}, doi = {10.30504/jims.2022.305123.1039}, abstract = {We introduced the $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and showed its monotone property, for example, $\mathfrak{R}^{[n]}_{x,x}(A|B) \le \mathfrak{R}^{[n]}_{x,y}(A|B) \le \mathfrak{R}^{[n]}_{y,y}(A|B)$ and $\mathfrak{R}^{[n]}_{x,x}(A|B) \le \mathfrak{R}^{[n]}_{y,x}(A|B) \le \mathfrak{R}^{[n]}_{y,y}(A|B)$ for $x\le y$ if $A\le B$ or $n$ is odd.The $n$-th residual relative operator entropy $\mathfrak{R}^{[n]}_{x,y}(A|B)$ is not symmetric on $x$ and $y$, that is, $\mathfrak{R}^{[n]}_{x,y}(A|B)\neq \mathfrak{R}^{[n]}_{y,x}(A|B)$ for $n\geq 2$ while $\mathfrak{R}^{[1]}_{x,y}(A|B)= \mathfrak{R}^{[1]}_{y,x}(A|B)$.In this paper we compare $\mathfrak{R}^{[n]}_{x,y}(A|B)$ with $\mathfrak{R}^{[n]}_{y,x}(A|B)$ and give the relations between $\mathfrak{R}^{[n]}_{x,y}(A|B)$ and the $n$-th operator divergence $\Delta_{i,x}^{[n]}(A|B)$.In this process, we find another operator divergence ${\overline \Delta}_{i,x}^{[n]}(A|B)$ which is similar to $\Delta_{i,x}^{[n]}(A|B)$ but not the same.}, keywords = {the n-th relative operator entropy,the n-th residual relative operator entropy,the n-th operator valued divergence}, url = {https://jims.ims.ir/article_144305.html}, eprint = {https://jims.ims.ir/article_144305_9e853377ecb86fb4bc9ccfcf7be09985.pdf} }