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Let A and 
$\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of 
$\tilde A$ lie, if A and 
$\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of 
$\tilde A$, assuming that 
$\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.
Let 
${\mathcal{H}}=\mathbb{C}^{n}\otimes {\mathcal{E}}$ be the tensor product of a Euclidean space 
$\mathbb{C}^{n}$ and a separable Hilbert space 
${\mathcal{E}}$. Our main object is the operator 
$G=I_{n}\otimes S+A\otimes I_{{\mathcal{E}}}$, where 
$S$ is a normal operator in 
${\mathcal{E}}$, 
$A$ is an 
$n\times n$ matrix, and 
$I_{n},I_{{\mathcal{E}}}$ are the unit operators in 
$\mathbb{C}^{n}$ and 
${\mathcal{E}}$, respectively. Numerous differential operators with constant matrix coefficients are examples of operator 
$G$. In the present paper we show that 
$G$ is similar to an operator 
$M=I_{n}\otimes S+\hat{D}\times I_{{\mathcal{E}}}$ where 
$\hat{D}$ is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of 
$G$, nonregular on the closed convex hull 
$\operatorname{co}(G)$ of the spectrum of 
$G$. The functions 
$G^{-\unicode[STIX]{x1D6FC}}\;(\unicode[STIX]{x1D6FC}>0)$ and 
$(\ln G)^{-1}$ are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of 
$G$ regular on 
$\operatorname{co}(G)$. Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.
Let 
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of 
$H$ is a compact operator and 
$H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that 
$H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.
