### Two models for contraction operators. (Deux modèles pour les opérateurs de contraction.)

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2000 Mathematics Subject Classification: 47A10, 47A13. In this paper, we give a description of Taylor spectrum of commuting 2-contractions in terms of characteritic functions of such contractions. The case of a single contraction obtained by B. Sz. Nagy and C. Foias is generalied in this work.

Let $\mathcal{B}\left(\mathscr{H}\right)$ be the set of all bounded linear operators acting in Hilbert space $\mathscr{H}$ and ${\mathcal{B}}^{+}\left(\mathscr{H}\right)$ the set of all positive selfadjoint elements of $\mathcal{B}\left(\mathscr{H}\right)$. The aim of this paper is to prove that for every finite sequence ${\left({A}_{i}\right)}_{i=1}^{n}$ of selfadjoint, commuting elements of ${\mathcal{B}}^{+}\left(\mathscr{H}\right)$ and every natural number $p\ge 1$, the inequality $$\frac{{e}^{p}}{{p}^{p}}\left(\sum _{i=1}^{n}{A}_{i}^{p}\right)\le exp\left(\sum _{i=1}^{n}{A}_{i}\right)\phantom{\rule{0.166667em}{0ex}},$$ holds.

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