Extremal case in Marcus-Oliveira conjecture and beyondстатья
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Дата последнего поиска статьи во внешних источниках: 18 декабря 2013 г.
Аннотация:For $A, C \in M_n$ the $C$-determinantal range of $A$ is the
following set on the complex plane
$\bigtriangleup_C(A) = \{\det(A-UCU^*):\, UU^*=I_n\}$. For normal
matrices $A$ and $C$ with eigenvalues $\al_1,\ldots,\al_n$ and
$\ga_1,\ldots,\ga_n$, respectively, Marcus \cite{Mar1} and Oliveira
\cite{GO} conjectured that $\bigtriangleup_C(A)$ is a subset of the
convex hull of
the points $z_\sigma=\prod_{j=1}^n (\al_j-\ga_{\sigma(j)})$,
$\sigma\in S_n$, where $S_n$ is the symmetric group of degree $n$.
We investigate the extremal set of matrices for
which the equality holds
in Marcus-Oliveira Conjecture.
We illustrate the use of the obtained results by two different
applications. The first one deals with the
equality case between the radius of $\bigtriangleup_C(A)$ and
the
radius of the convex hull of the points $z_\sigma$, $\sigma \in
S_n$.
The second one is the characterization of additive
Frobenius endomorphisms for the determinantal range or radius on
the space $M_n$ and on its real subspace of Hermitian matrices.