No abstract

ABSTRACT. In the space l~ of vector sequences, we consider the symmetric operator L generated by the expression (lu)j := Bjuj+l + Ajuj + B~_luj_l , where u-1 --0, u0, ut .... E C p, Aj and Bj are p x p matrices with entries from C, A~ = Aj, and the inverses B~ -1 (j = 0, 1,...) 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 I in terms of the coefficients Aj and Bj are derived.KEY WOADS: sequence space, difference expression, matrix polynomial, deficiency numbers of an operator.w Introduction Let C p (p _~ 1) be the p-dimensional Euclidean space of column vectors with the standard inner P product y*x = )"~1=1 xjyj, where xj and yj (j = 1, 2,...~ p) are the coordinates of the vectors x and y, respectively, and the asterisk indicates the adjoint matrix. Further, let Aj and Bj (j = 0, 1, ...) be p x p matrices with complex entries; we assume that Aj is a self-adjoint matrix and Bj is invertible (j = 0, 1,... ). Consider the second-order difference expressionwhere u-1 := 0, u0, ul, .-. E C p 9 Let the symbol I denote the operation that acts on sequences of p x p matrices U0, U1,... with entries from C in the same way as l, i.e., according to (1). The following Green formula is valid: ~t A similar expression is also valid for the case in which the vectors uj and vj are replaced by the matrices Uj and V~ and the operation 1 is replaced by I. ~-~((lu)iv I -ui(lv)j) = [uvl(s)[~_,,Let s be the Hilbert space of infinite sequences of vectors u = (uo, ul, ... ), u i E C v, with inner product (u, v) w,+oo . = 2-.,i=o vjuj, and let L be the minimal closed operator generated by the expression (1) with the boundary condition u-1 = 0 in the space s (see [1, Chap. VII, w p. 509;2]). It is well known that the operator L is symmetric but is not self-adjoint in general, and the deficiency numbers n+ and n_ of this operator satisfy the inequalities 0 < n+, n_ <_ p. We say that we have a completely indefinite case for the expression l if n+ = n_ --p. It is readily derived from the subsequent argument (see Theorem 1 and its proof) that we have the completely indefinite case for 1 if and only if all solutions of the equation (lu)j=zuj, j=0,1,..., u_,=0,for z = 0 belong to the space d~2p.

We study the deficiency numbers of Jacobi matrices with square matrix entries and of symmetric ordinary differential operators with polynomial coefficients in the spaces 12 and .~2(-=oo, +oo), respectively. Jacobi matrices with matrix entries are known to occur, say, in the matrix moment problem. In.particular, here we find a criterion for a Jacobi matrix with matrix entries of a special form to be completely indeterminate (see the d~nition below). It follows from this criterion that the operators generated by completely indeterminate Jacobi matrices in the space 12 are entire operators of the minimal type (see [1,2]). Consequently, selfadjoint extensions of these operators have discrete spectra, the spectral measures possess a certain structure, etaThe matrix representation of symmetric ordinary differential operators with polynomial coefficients defined on the entire real axis is another source of these Jacobi matrices. Thus, there occur new classes of entire operators of the minimal type with prescribed deficiency numbers. Furthermore, the deficiency numbers of these operators can be less than, or equal to, or, which is most interesting, greater than the order of the corresponding differential expression.Thus, the class of entire operators discovered by M. Krein [1, 2] turns out to be substantially wider than it was conceived earlier.w Preliminaries 0

No abstract