The symmetrization map and \Gamma-contractions
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Pal, Sourav [1]
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[1] Mathematics Department, Indian Institute of Technology Bombay, Powai, Mumbai, 400076, India
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- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 75, Fasc. 1, 2024, págs. 81-99
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
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, \begin{aligned} \Gamma = \pi (\overline{{\mathbb{D}}^2})=\{ (z_1+z_2,z_1z_2)\,:\, |z_i|\le 1, i=1,2 \}. \end{aligned} 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 \{ \Vert T_1\Vert ,\Vert T_2\Vert \} 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 \{\Vert A_1\Vert , \Vert A_2\Vert \} \le 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).
