Extremality of numerical radii of matrix products

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• 作者：Hwa-Long Gau^a ; ¹ ; ^{hlgau@math.ncu.edu.tw" class="auth_mail" title="E-mail the corresponding author} ; Pei Yuan Wu^b ; ^{pywu@math.nctu.edu.tw" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A60 ; 15A27
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 July 2016
• 年：2016
• 卷：501
• 期：Complete
• 页码：17-36
• 全文大小：420 K

文摘

For two n-by-n matrices A and B  , it was known before that their numerical radii satisfy the inequality w(AB)≤4w(A)w(B)$w(AB)\le 4w(A)w(B)$, and the equality is attained by the 2-by-2 matrices $A=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ and $B=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$. Moreover, the constant “4” here can be reduced to “2” if A and B   commute, and the corresponding equality is attained by $A=I_2\otimes \left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$ and $B=\left[\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\otimes I_2$. In this paper, we give a complete characterization of A and B   for which the equality holds in each case. More precisely, it is shown that w(AB)=4w(A)w(B)$w(AB)=4w(A)w(B)$ (resp., w(AB)=2w(A)w(B)$w(AB)=2w(A)w(B)$ for commuting A and B) if and only if either A or B is the zero matrix, or A and B   are simultaneously unitarily similar to matrices of the form $\left[\begin{array}{cc}\hfill 0\hfill & \hfill a\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\oplus A^{\prime }$ and $\left[\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill b\hfill & \hfill 0\hfill \end{array}\right]\oplus B^{\prime }$ (resp., with w(A^′)≤|a|/2$w(A^{\prime })\le |a|/2$ and w(B^′)≤|b|/2$w(B^{\prime })\le |b|/2$. An analogous characterization for the extremal equality for tensor products is also proven. For doubly commuting matrices, we use their unitary similarity model to obtain the corresponding result. For commuting 2-by-2 matrices A and B  , we show that w(AB)=w(A)w(B)$w(AB)=w(A)w(B)$ if and only if either A or B is a scalar matrix, or A and B   are simultaneously unitarily similar to $\left[\begin{array}{cc}\hfill a_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill a_2\hfill \end{array}\right]$ and $\left[\begin{array}{cc}\hfill b_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill b_2\hfill \end{array}\right]$ with |a₁|≥|a₂|$|a_1|\ge |a_2|$ and |b₁|≥|b₂|$|b_1|\ge |b_2|$.