Estimates of the spectra and the invertibility of functional operators

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.