Three-term recurrence relations with matrix coefficients. The completely indefinite case

In the space $\ell_p^2$ of vector sequences, we consider the symmetric operator $L$ generated by the expression $(lu)_j:=B_ju_{j+1}+A_ju_j+B_{j-1}^*u_{j-1}$, where $u_{-1}=0$, $u_0,u_1,\ldots\in\mathbb C^p$, $A_j$ and $B_j$ are $p\times p$ matrices with entries from $\mathbb C$, $A_j^*=A_j$, and the inverses $B_j^{-1}$ ($j=0,1,\dots$) exist. We state a necessary and sufficient condition for the deficiency numbers of the operator $L$ to be maximal; this corresponds to the completely indefinite case for the expression $l$. Tests for incomplete indefiniteness and complete indefiniteness for $l$ in terms of the coefficients $A_j$ and $B_j$ are derived.