Abstract
In the space {ol 2p of vector sequences, we consider the symmetric operatorL generated by the expression (lu)j:=Bj uj+1+Aj uj+ j−1/* B uj−1, whereu−1 = 0,u 0,u 1, … ∈ ℂp,A j andB j arep × p matrices with entries from ℂ,A *j =Aj, and the inversesB −1 j (j = 0, 1, …) exist. We state a necessary and sufficient condition for the deficiency numbers of the operatorL to be maximal; this corresponds to the completely indefinite case for the expressionl. Tests for incomplete indefiniteness and complete indefiniteness forl in terms of the coefficientsA j andB j are derived.
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Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 709–716, May, 1998.
This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00333.
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Kostyuchenko, A.G., Mirzoev, K.A. Three-term recurrence relations with matrix coefficients. The completely indefinite case. Math Notes 63, 624–630 (1998). https://doi.org/10.1007/BF02312843
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DOI: https://doi.org/10.1007/BF02312843