Deficiency numbers of symmetric operators generated by block Jacobi matrices

Symmetric block Jacobi matrices $J$ and the corresponding symmetric operators $L$ are studied. Let $m$ be the size of the blocks in the matrix $J$. As is known, the deficiency numbers $m_+$ and $m_-$ of the operator $L$ satisfy the inequalities $0\leqslant m_+,m_-\leqslant m$ and achieve their maximum value $m$ simultaneously. Let $m_+$ and $m_-$ be arbitrary integers such that $0\leqslant m_+,m_-\leqslant m-1$. It is shown that there exists a symmetric Jacobi matrix $J$ such that $m_+$ and $m_-$ are the deficiency numbers of the corresponding symmetric operator $L$.
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