数理解析研究所講究録
第 1596 巻 2008 年 140-142140

行列環のテンソル積空間上の部分転置写像について

山形大学・理学部数理科学科 佐野隆志 (Takashi SANO)
Department of Mathematical Sciences
Faculty of Science, Yamagata University

This is a survey of [1], which is a joint work with Tsuyoshi Ando.
Here are some of the results in [1]. For more details, we refer the reader
to [1].
1. 部分転置写像とその関連
Theorem 1.1. Let $A_{k}\in M_{m},$ $B_{k}\in M_{n}$ . Then for every unitarily
invariant norm$|||\cdot|||$$\sup_{X\in M_{m_{\backslash }\mathfrak{n}}}\frac{|||\sum_{k}A_{k}XB_{k}^{T}|||}{|||X|||}\leqq\sqrt{\min(m,n)}\Vert\sum_{k}A_{k}\otimes B_{k}\Vert_{\infty}$Theorem 1.2. Let $A_{k}\in M_{m},$invariant norm

$B_{k}\in M_{n}$.

. Then for every unitarily$|||\cdot|||$$\sup_{X\in M_{m,n}}\frac{|||\sum_{k}A_{k}XB_{k}|||}{|||X|||}\leqq$

The partial transpose map$\Theta$min $(m, n) \Vert\sum_{k}A_{k}\otimes B_{k}\Vert_{\infty}$ .

on$M_{m}\otimes M_{n}$is defined as$\Theta(\sum_{k}A_{k}\otimes B_{k})=\sum_{k}A_{k}\otimes B_{k}^{T}$.

Theorem 1.3. For every unitarily invariant norm Ill$|||$$||| \Theta(X)|||\leqq\min(m, n)|||X|||$ .

For the Hilbert-Schmidt norm$||\cdot||_{2}$$\Vert\Theta(X)||_{2}=\Vert X\Vert_{2}$.

Notice that this inequality for the spectral norm was originally established by Tomiyama

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2. 部分転置写像の制限

Let$K_{m,n}$$:=\{A\otimes I_{n}+I_{m}\otimes B:A\in M_{m}, B\in M_{n}\}$

We observe the restriction$\Theta_{0}$of the map$\Theta_{0}(X)=A\otimes I_{n}+I_{m}\otimes B^{T}$

Theorem 2.1. The rest

$r\dot{\tau}ction\Theta_{0}$$\Theta$.

to the subspace $K_{m,n}$ :for $X=A\otimes I_{n}+I_{m}\otimes B$ .
is positive:$x\geqq 0\Rightarrow\Theta_{0}(X)\geqq 0$.

Theorem 2.2. Let $X=A\otimes I_{n}+I_{m}\otimes B$ .
(i) If $n=2$ or $B$ is no $rmal_{J}$ then X and$\Theta_{0}(X)$in.
(ii) If $m=2$ or $A$ is normal, then X andin.

are unitari$ly$similar$M_{m}\otimes M_{n}$$\Theta_{0}(X)^{T}$

are unitarily similar$M_{m}\otimes M_{n}$In any case, $|X|$ and;are unitarily similar inhence, $|||\Theta_{0}(X)|||=|||X|||$ for every
invariant norm II1 . .
Theorem 2.3. If $X\in K_{m,n}$ is normal, then X and
are unitarily$|||X|||$similar andevery
invariant no $rm$for.
Theorem 2.4. For every unitari inva riant norm . Ill$M_{m}\otimes M_{n}$$|\Theta_{0}(X)|$$unitari\cdot ly$$|||$$\Theta_{0}(X)$$|||\Theta_{0}(X)|||$$unita7\dot{\tau}ly$$=$$|||\cdot|||$$ly$$|||\Theta_{0}(X)|||\leqq 2|||X|||$$|||$$(X\in K_{m,n})$.

We recall the definition of the numerical mnge $W(X)$ of a squarematrix $X\in M_{n}$ :$W(X):=\{\langle Xa, a\rangle :a\in \mathbb{C}^{n}, \Vert a\Vert=1\}$

The numerical radius $w(X)$ is defined as{ I : $\xi\in W(X)$ }Theorem 2.5 . For $X\in K_{m,n}$$w(X)= \sup$$|\xi$$= \sup\{|\langle Xa, a\rangle| :a\in \mathbb{C}^{n}, ||a\Vert=1\}$$\cdot$$w(\Theta_{0}(X))=w(X)$and more precisely,$W(\Theta_{0}(X))=W(X)$ .

Theorem 2.6. Let $A\in M_{m},$ $B\in M_{n}$ . Then for every unitarily invanant no$\tau m|||\cdot|||$$\sup_{X\in M_{m,n}}\frac{|||AX+XB^{T}|||}{|||X|||}\leqq\sqrt{2}\Vert A\otimes I_{n}+I_{m}\otimes B\Vert_{\infty}$and$\sup_{X\in M_{m,\mathfrak{n}}}\frac{|||AX+XB|||}{|||X|||}\leqq\sqrt{2}\Vert A\otimes I_{n}+I_{m}\otimes B\Vert_{\infty}$.

,

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Theorem 2.7. For $A\in M_{m}$ and $B\in M_{n}$$\Vert A\otimes I_{m}+I_{n}\otimes B^{T}\Vert_{\infty}\leqq\sqrt{2}\Vert A\otimes I_{n}+I_{m}\otimes B\Vert_{\infty}$

Recall the definition of the Schatten p-norm$\Vert X\Vert_{p}$$\Vert$$:= \{\sum_{i}s_{i}(X)^{p}\}^{1/p}$.

. lip for $1\leqq p<\infty$ :.

Theorem 2.8. For $p=2k(k=1,2,3,4,5)$$\Vert\Theta_{0}(X)\Vert_{p}=\Vert X\Vert_{p}$$(X\in K_{m,n})$.

REFERENCES[1] T. Ando and T. Sano, Norm estimates of the partial transpose map on thetensor products of matrices, Positivity, 12 (2008), 9-24.