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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]lpawela Both sides previous revision Previous revision 2013/05/08 12:56 lpawela 2013/05/08 12:45 lpawela 2013/05/08 12:31 lpawela 2013/05/08 12:31 lpawela 2013/05/08 12:30 lpawela 2013/05/08 12:18 lpawela created 2013/05/08 12:56 lpawela 2013/05/08 12:45 lpawela 2013/05/08 12:31 lpawela 2013/05/08 12:31 lpawela 2013/05/08 12:30 lpawela 2013/05/08 12:18 lpawela created Line 9: Line 9: $\Lambda_k(M)$ 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 $$$Line 18: Line 18: 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.$$$