Geodesic
Estimates of the spectra and the invertibility of functional operators
Sbornik. Mathematics, Tome 34 (1978) no. 2, pp. 243-258 Cet article a éte moissonné depuis la source Math-Net.Ru

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For an $\mathfrak R$-valued function $f(z_1,\dots,z_n)$ ($\mathfrak R$ is a Banach algebra) that is holomorphic in a neighborhood $\Omega$ of the joint spectrum of $n$ elements $B_1,\dots,B_n\in\mathfrak R$ that commute with each other and with $f(z_1,\dots,z_n)$ $\forall\,z=(z_1,\dots,z_n)\in\Omega$, the function $f(B_1,\dots,B_n)$ is introduced and estimates of the spectrum $\sigma(f(B_1,\dots,B_n))$ are given, one of which generalizes the maximum principle for holomorphic functions. The estimates of $\sigma(f(B_1,\dots,B_n))$ are used to solve problems on the invertibility of transformers, operators induced by discrete systems and operators induced by linear differential equations with constant deviations of the argument. Bibliography: 11 titles. 
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     title = {Estimates of the spectra and the invertibility of functional operators},
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V. E. Slyusarchuk. Estimates of the spectra and the invertibility of functional operators. Sbornik. Mathematics, Tome 34 (1978) no. 2, pp. 243-258. http://geodesic.mathdoc.fr/item/SM_1978_34_2_a7/

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