Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The symmetrization map and $\Gamma$-contractions   The symmetrization map $\pi:\mathbb C^2\rightarrow \mathbb C^2$ is defined by
$ \pi(z_1,z_2)=(z_1+z_2,z_1z_2). $ The closed symmetrized bidisc $\Gamma$ is
the symmetrization of the closed unit bidisc $\overline{\mathbb D^2}$, that is,
\[ \Gamma = \pi(\overline{\mathbb D^2})=\{ (z_1+z_2,z_1z_2)\,:\, |z_i|\leq 1,
i=1,2 \}. \] A pair of commuting Hilbert space operators $(S,P)$ for which
$\Gamma$ is a spectral set is called a $\Gamma$-contraction. Unlike the scalars
in $\Gamma$, a $\Gamma$-contraction may not arise as a symmetrization of a pair
of commuting contractions, even not as a symmetrization of a pair of commuting
bounded operators. We characterize all $\Gamma$-contractions which are
symmetrization of pairs of commuting contractions. We show by constructing a
family of examples that even if a $\Gamma$-contraction $(S,P)=(T_1+T_2,T_1T_2)$
for a pair of commuting bounded operators $T_1,T_2$, no real number less than
$2$ can be a bound for the set $\{ \|T_1\|,\|T_2\| \}$ in general. Then we
prove that every $\Gamma$-contraction $(S,P)$ is the restriction of a
$\Gamma$-contraction $(\widetilde S, \widetilde P)$ to a common reducing
subspace of $\widetilde S, \widetilde P$ and that $(\widetilde S, \widetilde
P)=(A_1+A_2,A_1A_2)$ for a pair of commuting operators $A_1,A_2$ with $\max
\{\|A_1\|, \|A_2\|\} \leq 2$. We find new characterizations for the
$\Gamma$-unitaries and describe the distinguished boundary of $\Gamma$ in a
different way. We also show some interplay between the fundamental operators of
two $\Gamma$-contractions $(S,P)$ and $(S_1,P)$.

23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: econ
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
31
The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: cs
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an