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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 12:31]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 Next 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 Line 5: Line 5: as a tool to handle compression problems in quantum information theory. Since then as a tool to handle compression problems in quantum information theory. Since then their theory and applications have been advanced with remarkable enthusiasm. The their theory and applications have been advanced with remarkable enthusiasm. The - sequence of papers ​ADD CITE!, for example, led to a striking + sequence of papers ​[( :​choi2007higher )], [( :​choi2008geometry )], [( :​woerdeman2008higher )], [( :​li2008canonical )], for example, led to a striking extension of the classical Toeplitz--Hausdorff theorem (convexity of $W(M)$): **all** the extension of the classical Toeplitz--Hausdorff theorem (convexity of $W(M)$): **all** the $\Lambda_k(M)$ 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.$$$