Abstract: | In the space {ol
p
2
of vector sequences, we consider the symmetric operatorL generated by the expression (lu)j:=Bj
uj+1+Aj
uj+
B
j−1/*
uj−1, whereu−1 = 0,u
0,u
1, … ∈ ℂ
p
,A
j andB
j arep × p matrices with entries from ℂ,A
j
*
=Aj, and the inversesB
j
−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 coefficientsA
j andB
j are derived.
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. |