J -self-adjointness of a class of Dirac-type operators

In this note, we prove that the maximally defined operator associated with the Dirac-type differential expression M(Q) = i d dx Im −Q −Q∗ − d dx Im , where Q represents a symmetric m × m matrix (i.e., Q(x) = Q(x) a.e.) with entries in L1 loc(R), is J-self-adjoint, where J is the antilinear conjugation defined by J = σ1C, σ1 = 0 Im Im 0 and C(a1, . . ., am,b1, . . ., bm) = (a1, . . . , am, b1, . . . , bm) . The differential expression M(Q) is of significance as it appears in the Lax formulation of the non-abelian (matrix-valued) focusing nonlinear Schrödinger hierarchy of evolution equations.  2004 Elsevier Inc. All rights reserved.