$$\left(B-\lambda I\right)^{-1}=\left(\begin{array}{cc} \frac{1}{1-\lambda} & -\frac{1}{(1-\lambda)^2}\\ 0 & \frac{1}{1-\lambda} \end{array} \right),\: \lambda\neq 1.$$

This observation partially confirms the following commonly known thought: the spectrum does not contain much information about the behaviour of NSA operator $A$, see also [@Davies1], [@Davies2]. For this reason the notion of pseudospectra was introduced and came into play. The pseudospectra of $A$ is the family of sets $$\left\{\lambda\in\mathbb{C}:\: \left\| \left(A-\lambda I\right)^{-1}\right\|  \geq \frac{1}{\varepsilon}\right\}_{\varepsilon>0},$$ see [@Davies1], [@Davies2], and it describes the behaviour of NSA operator $A$ much more effectively than the spectrum.

Another way to control the resolvent is to obtain for it estimates from above, see [@Davies1], [@Davies3], and works [@Malejki], [@Eisner1], [@Eisner2], [@Eisner3], where direct links between the polynomial growth of the $C_0$-semigroup $\left\{e^{At} \right\}_{t\geq 0}$ in $t$ and the behaviour of the corresponding resolvent $\left(A-\lambda I\right)^{-1}$ were established. Note that $C_0$-semigroups and $C_0$-groups with polynomial growth condition naturally appear in theory and applications of evolution equations, see, e.g., [@Goldstein], [@Sklyar2].

In 1985 A.I. Miloslavskii [@Miloslavskii] obtained sufficient conditions for $C_0$-semigroup on a Hilbert space to be polynomially bounded in terms of the behaviour of eigenvalues of the corresponding generator, under the assumption (\[question\]). B.A. Barnes [@Barnes] in 1989 obtained a number of interesting properties 