On local invertible operators in L-2(R-1, H)

We study operators of the form Lu = d(2)u/dt(2) - G(t)u(t) in L-2 ([t(o) - delta, t(o) + delta], H) with <(D(L))over bar> = L-2([t(o) - delta, t(o) delta], H) in the neighbourhood [t(o) - delta, t(o) + delta] of a point t(o ) is an element of R-1. Such problems arise in questions on local solvabili ty of partial differential equations (see [6] and [7]). For these operators , one of the major questions is if they are invertible in a neighbourhood o f a point t is an element of R-1. To solve this problem we establish needed commutator estimates. Using the commutator estimates and factorization the orems for nonanalytic operator-functions we give additional conditions for the nonanalytic operator-function G(t) and show that the operator L (or (L) over bar) with some boundary conditions is local invertible.