Riesz Basis of Eigenvectors for Analytic Families of Operators and Application to a Non-symmetrical Gribov Operator

In the present paper, we are mainly concerned with the existence of a Riesz basis related to the Gribov operator

$$\begin{aligned} A^{*2}A^2+\varepsilon (A^* A + A^* (A + A^* )A), \end{aligned}$$

where \(\varepsilon \in \mathbb {C}\); while A is the annihilation operator and \(A^*\) is the creation operator verifying \(A, A^*] = I.\) Through a specific growing inequality, we extend this problem to a theoretical one and we study the invariance of the closure, the comportment of the spectrum as well as the existence of Riesz basis of generalized eigenvectors.