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numerical-range:generalizations:higher-order-numerical-range

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numerical-range:generalizations:higher-order-numerical-range [2013/05/08 12:45]
lpawela
numerical-range:generalizations:higher-order-numerical-range [2013/05/08 12:56] (current)
lpawela
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 $\Lambda_k(M)$ are convex (though some may be empty), and they are intersections $\Lambda_k(M)$ are convex (though some may be empty), and they are intersections
 of conveniently computable half--planes in $\mathbb{C}$. of conveniently computable half--planes in $\mathbb{C}$.
-Among the many more recent papers concerning the $\Lambda_k(M)$,​ let us mention ​CITE!! ​[LPS2009,​GLW2010].+Among the many more recent papers concerning the $\Lambda_k(M)$,​ let us mention [( :​li2009condition )] and [( :​gau2010higher )].
  
-Given a matrix $M\in M_N$ and $k\geq1$, Choi, Kribs, and \.Zyczkowski ​(see [CK\.Z2006a,​CK\.Z2006b])+Given a matrix $M\in M_N$ and $k\geq1$, Choi, Kribs, and Życzkowski ​(see [( :​choi2006higher )])
 defined the rank--$k$ numerical range of $M$ as defined the rank--$k$ numerical range of $M$ as
 \[ \[
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 where $P_k$ denotes the set of rank--$k$ orthogonal projections in $M_N$. It is not hard where $P_k$ denotes the set of rank--$k$ orthogonal projections in $M_N$. It is not hard
 to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that to verify that $\Lambda_K(M)$ can also be described as the set of complex $\lambda$ such that
-there is some $k$--dimensional ​ subspace $S$ of $\mathbb{C}^N$ such that $(Mu,​u)=\lambda$ for \textbf{all}+there is some $k$--dimensional ​ subspace $S$ of $\mathbb{C}^N$ such that $(Mu,​u)=\lambda$ for **all**
 unit vectors in $S$. In particular, we see that unit vectors in $S$. In particular, we see that
 \[ \[
 W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad. W(M)=\Lambda_1(M)\supseteq\Lambda_2(M)\supseteq\Lambda_3(M)\supseteq\dots\quad.
 \] \]
numerical-range/generalizations/higher-order-numerical-range.txt · Last modified: 2013/05/08 12:56 by lpawela