\[q\] in the case when $A$ is a perturbation of selfadjoint, nonnegative operator with discrete spectrum, including perturbations of harmonic oscillator type operators. We refer to [@Mityagin], Section 4.3, for the brief overview of the corresponding results.

In the study of (in fact, quite old) Question \[q\] a breakthrough was made by G.Q. Xu, S.P. Yung and H. Zwart – the XYZ Theorem:

If the following three conditions hold:

1.  The operator $A$ generates the $C_0$-group on a Hilbert space $H$;

2.  The eigenvalues $\{\lambda_n\}_{n=1}^{\infty}$ of $A$ is a union of $K<\infty$ interpolation sequences $\Lambda_k$, $1\leq k\leq K.$ In other words, $\{\lambda_n\}_{n=1}^{\infty}=\bigcup\limits_{k=1}^K \Lambda_k$, where $$\label{1}
     \min_{k}\inf\limits_{\lambda_{n},\lambda_{m}\in \Lambda_k:\: n\neq m} |\lambda_{n}-\lambda_{m}|>0;$$

3.  The span of the generalized eigenvectors (eigen- and rootvectors) of $A$ is dense,

then the condition (\[question\]) holds.

More precisely [@Zwart], under the three conditions above, there exists a certain sequence of (multidimensional, if $K>1$) spectral projections $\{P_n\}_{n=1}^{\infty}$ of $A$ such that $\{P_n H\}_{n=1}^{\infty}$ forms a Riesz basis of subspaces in $H$ with $$\sup\limits_{n\in\mathbb{N}} \dim P_n H\leq K.$$

Operators satisfying conditions 1-3 of the XYZ Theorem naturally arise from applications, e.g., in the analysis of neutral type systems [@Rabah1], [@Rabah2], [@Sklyar1], [@Sklyar2]. Since it is usually not hard to 