On root elements of an operator matrix

Let $H$ be a Hilbert space and let ${A:\mathcal D(A)\subset H\rightarrow H}$ be a selfadjoint positive definite operator, ${A^{-1}\in\mathfrak S_q(H)}$ ${(q>0)}$, ${\beta_l>0}$ ${(l=\overline{0,m})}$, ${0=:b_0<b_1<\ldots<b_m}$.

Define ${\mathcal H:=H\oplus\big(\oplus_{l=0}^mH\big)}$. The Hilbert space $\mathcal H$ consists of elements of the form ${\xi:=(v;w)^{\tau}:=\big(v;(v_0;v_1;\dots;v_m)^{\tau}\big)^{\tau}}$. Let an operator $\mathcal A$ be given by the following formulae:
\begin{align*} &\qquad\qquad \mathcal A={\rm{diag}}(A^{1/2},\mathcal I) \left(\!\!\! \begin{array}{cc} 0 & \mathcal Q^*\\ -\mathcal Q & \mathcal G \\ \end{array}\!\!\!\right) {\rm{diag}}(A^{1/2},\mathcal I), \\ &\mathcal Q:=\big(\beta_0^{1/2}I,\beta_1^{1/2}I,\dots, \beta_m^{1/2}I\big)^{\tau},\quad \mathcal G:={\rm{diag}}\big(0,b_1I,\dots, b_mI\big), \nonumber\\ &\mathcal D(\mathcal A)=\Big\{\xi\in\mathcal H\big\vert\; v\in\mathcal D(A^{1/2}),\;\; \mathcal Q^*w=\sum_{l=0}^m\beta_l^{1/2}v_l\in\mathcal D(A^{1/2})\Big\}.\nonumber \end{align*}

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
\begin{equation*} \begin{split} g_k(\lambda)&:=\mathcal Q^*(\mathcal G-\lambda)^{-1}\mathcal Q-\lambda\lambda_k\equiv-\frac{1}{\lambda}\beta_0+\sum_{l=1}^m\frac{\beta_l}{b_l-\lambda}-\lambda\lambda_k, \quad k\in\mathbb N,\\ g_\infty(\lambda)&:=\mathcal Q^*(\mathcal G-\lambda)^{-1}\mathcal Q\equiv-\frac{1}{\lambda}\beta_0+\sum_{l=1}^m\frac{\beta_l}{b_l-\lambda}\equiv-\frac{1}{\lambda} \Big[\sum_{l=0}^m\beta_l-\sum_{l=1}^m\frac{\beta_lb_l}{b_l-\lambda}\Big]. \end{split} \end{equation*}
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 ${g_{k}(\lambda)=0}$ ${(k\in\mathbb N)}$.

In non-degenerate case we prove the following theorem.

Theorem. Suppose that ${g^{\prime}_k(\lambda_k^{(p)})\ne0}$ ${(p=\overline{1,m+2},\;k\in\mathbb N)}$. Then the system ${\{\xi_{k}^{(p)}\}_{p=\overline{1,m+2},\;k\in\mathbb N}}$ of eigenelements of the operator $\mathcal A$ is defined by the following formulae
\begin{equation*} \begin{split} \xi_{k}^{(p)}:=&R_{k,p}\big(\lambda_k^{1/2};(\mathcal G-\lambda_k^{(p)})^{-1}\mathcal Q\big)^{\tau}u_k,\quad p=\overline{1,m+2}, \quad k\in\mathbb N,\\ &R_{k,p}:= \begin{cases} \big[g^{\prime}_{\infty}(\gamma_p)\big]^{-1/2},\quad p=\overline{1,m},\quad k\in\mathbb N,\\ \big[2\lambda_k\big]^{-1/2}, \quad p=m+1,m+2,\quad k\in\mathbb N \end{cases} \end{split} \end{equation*}
and forms a $p$-basis ${(p\geqslant 2q)}$ in the Hilbert space $\mathcal H$. The biorthogonal system has the form
\begin{equation*} \zeta_k^{(p)}:=-\big[g^{\prime}_k(\overline{\lambda_k^{(p)}})R_{k,p}\big]^{-1} \big(\lambda_k^{1/2};-(\mathcal G-\overline{\lambda_k^{(p)}})^{-1}\mathcal Q\big)^{\tau} u_k,\quad p=\overline{1,m+2}, \quad k\in\mathbb N. \end{equation*}

In degenerate case we prove that the system of the root elements of the operator $\mathcal A$ also forms a $p$-basis ${(p\geqslant 2q)}$ in the Hilbert space $\mathcal H$.