Norm Estimates of the Partial Transpose Map on the Tensor Products of Matrices

We present a norm estimate for the partial transpose map Θ on the tensor product \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_m \otimes M_n$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta \left( \sum_k A_k \otimes B_k \right) := \sum_k A_k \otimes B_k^T$$\end{document}with respect to a unitarily invariant norm. This is related to the norm estimates of the following maps on Mm,n in terms of the spectral norm of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\displaystyle \sum_k A_k \otimes B_k$$\end{document} : \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \longmapsto \sum_k A_k X B_k \quad \textrm{and} \quad X \longmapsto \sum_k A_k X B_k^T.$$\end{document}We show further that in the special case of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A \otimes I_n + I_m \otimes B$$\end{document} as well as AX + XB and AX + XBT those estimates are much improved and that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\| A \otimes I_n + I_m \otimes B^T \|_p = \| A \otimes I_n + I_m \otimes B \|_p$$\end{document}for certain Schatten p-norms.