  Let $A_j,B_j$ $(j=0,1,\ldots)$ be $m \times m$ matrices, whose elements are
complex numbers, $A_j$ are selfadjoint matrices and $B_j^{-1}$ exist. We study
the deficiency index problem for minimal closed symmetric operator $L$ with
domain $D_L$, generated by the Jacobi matrix $\textbf{J}$ with entries
$A_j,B_j$ in the Hilbert space $l_m^2$ of sequences $ u=(u_0,u_1, \ldots), u_j
\in C^m$ by mapping $u \rightarrow \textbf{J}u$, i.e. by the formula $Lu=lu$
for $u \in D_L$, where $lu=((lu)_0,(lu)_1, \ldots)$ and $$
(lu)_0:=A_0u_0+B_0u_1, \quad (lu)_j:=B^*_{j-1}u_{j-1}+A_ju_j+B_ju_{j+1}, \;\;
j=1,2, \ldots $$ It is well known that the case of the minimal deficiency
numbers of the operator $L$ corresponds to the determinate case, and the case
of the maximal deficiency numbers of this operator corresponds to the
completely indeterminate case of the matrix power moment problem. In this paper
we obtain new conditions of the minimal, maximal and not maximal deficiency
numbers of the operator $L$ in terms of the entries of the matrix $\textbf{J}$.
The special attention is paid to the case $m=1$, i.e. we present some
conditions on the elements of the numerical tridiagonal Jacobi matrix, which
ensure the realization of the determinate case of the classical power moment
problem.
