On the Basis Property for the Root Vectors of Some Nonselfadjoint Operators*

M here and in the sequel denotes various constants, and 1 ?“I the norm of operator T on H. Under assumptions (1 ), (2) the operator A = L + T has a discrete spectrum, that is, every point of its spectrum is an eigenvalue of tinite algebraic multiplicity. If 1 is an eigenvalue of A, then the linear hull of the corresponding eigenvectors is called the eigenspace corresponding to d. Let h, be an eigenvector, Ahj = 13.hj. If the equation Ah,!” = A@” + hj is solvable then the chain {h. /z(i) ,..., hjSj)}, AhfSj) = A/z,!~‘) + /zjSj-lF is called the Jordan chain correspondi& ;o the pair (A, hj). The number sj + 1 is called the length of this chain if the equation Ah Ah = h, tS~) has no solutions. If 1 has a finite algebraic multiplicity then sj < 00. The vectors hjrn) are called root vectors (or associated vectors). The union of eigen and root vectors is called the root system of A. A system { gj}j”, , of vectors is called linearly independent if any