Karakhan Agahan ogly Mirzoev

Three-Term Recurrence Relations With Matrix Coefficients. The Completely Indefinite Case

In the space {olp2 of vector sequences, we consider the symmetric operatorL generated by the expression (lu)j:=Bjuj+1+Ajuj+Bj−1/*uj−1, whereu−1 = 0,u0,u1, … ∈ ℂp,Aj andBj arep × p matrices with entries from ℂ,Aj*=Aj, and the inversesBj−1 (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 coefficientsAj andBj are derived.