Geodesic
Three-term recurrence relations with matrix coefficients. The completely indefinite case
Matematičeskie zametki, Tome 63 (1998) no. 5, pp. 709-716.

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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. 
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     author = {A. G. Kostyuchenko and K. A. Mirzoev},
     title = {Three-term recurrence relations with matrix coefficients. {The} completely indefinite case},
     journal = {Matemati\v{c}eskie zametki},
     pages = {709--716},
     publisher = {mathdoc},
     volume = {63},
     number = {5},
     year = {1998},
     language = {ru},
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A. G. Kostyuchenko; K. A. Mirzoev. Three-term recurrence relations with matrix coefficients. The completely indefinite case. Matematičeskie zametki, Tome 63 (1998) no. 5, pp. 709-716. http://geodesic.mathdoc.fr/item/MZM_1998_63_5_a8/

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