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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 $ℬ\left(\mathscr{H}\right)$ be the set of all bounded linear operators acting in Hilbert space $\mathscr{H}$ and ${ℬ}^{+}\left(\mathscr{H}\right)$ the set of all positive selfadjoint elements of $ℬ\left(\mathscr{H}\right)$. 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 ${ℬ}^{+}\left(\mathscr{H}\right)$ and every natural number $p\ge 1$, the inequality $$\frac{e^p}{p^p}\left(\sum _{i=1}^n{A}_i^p\right)\le exp\left(\sum _{i=1}^n{A}_i\right),$$ holds.