On root elements of an operator matrix.

Let $H$ be a Hilbert space and let $A: \mathscr{D}(A) \subset H \to H$ be a selfadjoint positive definite operator, $A^{-1} \in \mathscr{G}^q(H)\ (q>0),\ \beta_l \gt 0\ (l = \overline{0, m}), \ 0 = : b_0 \lt b_1 \lt \cdots \lt b_m$.
Define $\mathscr{H} := H \oplus (\oplus_{l = 0}^m H)$. The Hilbert spase $\mathscr{H}$ consist of elements of the form $\xi := (v; w)^\tau := (v; (v_0; v_1; \cdots; v_m)^\tau)^\tau$. Let an operator $\mathscr{A}$ be given by the following formulae: $$\mathscr{A} = diag (A^{1/2}, \mathscr{I}) \begin{pmatrix} 0 & \mathscr{Q}^\star \\ -\mathscr{Q} & \mathscr{G} \end{pmatrix} diag(A^{1/2}, \mathscr{I}),$$ $$\mathscr{Q} := (\beta_0^{1/2}I, \beta_1^{1/2}I, \cdots, \beta_m^{1/2}I)^\tau, \ \mathscr{Q} := diag(0, b_1I, \cdots, b_mI),$$ $$\mathscr{D}(\mathscr{A}) = \Bigr\{ \xi \in \mathscr{H}| v \in \mathscr{D}(A^{1/2}), \mathscr{Q}^\star w = \sum\limits_{l=0}^m \beta_l^{1/2}v_l \in \mathscr{D}(A^{1/2})\Bigl\}.$$
Let us denote by $\lambda_k = \lambda_k(A^{-1})$ and $u_k = u_k(A^{-1})$ $k \in \mathbb{N}$ the $k$-th eigenvalue and corresponding eigenelement of the operator $A^{-1}$ (i.e. the system $\{u_k\}_{k = 1}^\infty$ is an orthonormal basis of the Hilbert space $H$).
Let $g_k(\lambda)$ and $g_\infty(\lambda)$ be given by $$g_k(\lambda):= \mathscr{Q}^\star(\mathscr{G} - \lambda)^{-1}\mathscr{Q} - \lambda\lambda_k \equiv -\frac{1}{\lambda}\beta_0 + \sum\limits_{l=1}^m \frac{\beta_l}{b_l - \lambda} - \lambda\lambda_k, \ k \in \mathbb{N},$$ $$g_\infty(\lambda):= \mathscr{Q}^\star(\mathscr{G} - \lambda)^{-1}\mathscr{Q} \equiv -\frac{1}{\lambda}\beta_0 + \sum\limits_{l=1}^m \frac{\beta_l}{b_l - \lambda} \equiv -\frac{1}{\lambda}\biggl[\sum\limits_{l=0}^m\beta_l - \sum\limits_{l=1}^m\frac{\beta_lb_l}{b_l - \lambda}\biggr].$$
Let us denote by $\gamma_p \ (p=\overline{1,m})$ the roots of the equation $g_\infty(\lambda) = 0$. Let $\lambda_k^{(p)} \ (p = \overline{1, m + 2} )$ denote the roots of the equation $d_k(\lambda) = 0 \ (k \in \mathbb{N}).$
In non-degenerate case we prove the following theorem.
Theorem Suppose that $g_k^\prime(\lambda_k^{(p)}) \neq 0\ (p = \overline{1,m+ 2}, k \in \mathbb{N})$. Then the system $\{\xi_k^{(p)}\}_{p=\overline{1,m+2}, \ k \in \mathbb{N}}$ of eigenelements of the operator $\mathscr{A}$ is defined by the following formulae $$\xi_k^{(p)} := R_{k, p}(\lambda_k^{1/2}; (\mathscr{G} - \lambda_k^{(p)})^{-1} \mathscr{Q})^\tau u_k, \ p = \overline{1, m+2}, \ k \in \mathbb{N},$$ $$
R_{k,p} :=
\begin{cases}
[g_\infty^\prime(\gamma_p)]^{-1/2}, \ p=\overline{1,m}, \ k \in \mathbb{N}, \\
[2\lambda_k]^{-1/2}, \ p = m+1, m+2, \ k \in \mathbb{N}
\end{cases}
$$ and forms a p-basis $(p \ge 2q)$ in the Hilbert space $\mathscr{H}$. The biorthogonal system has the form $$
\zeta_k^{(p)} := - [g_k^\prime(\overline{\lambda_k^{(p)}})R_{k,p}]^{-1}(\lambda_k^{1/2}; -(\mathscr{G} - \overline{\lambda_k^{(p)}})^{-1} \mathscr{Q})^\tau u_k, \ p = \overline{1, m+2}, \ k \in \mathbb{N}.
$$ In degenerate case we prove that the system of the root elements of the operator $\mathscr{A}$ also forms a p-basis $(p \ge 2q)$ in the Hilbert space $\mathscr{H}$.
Keywords: operator matrix, spectrum, root element, basis, biorthogonal system.