J-selfadjoint ordinary differential operators similar to selfadjoint operators
by
Ilia Karabash
Donetsk State University, Donetsk, Ukraine

We consider the nonselfadjoint operators A_p = ( sgn x ) p( -i[ d/dx] ) in the Hilbert space L² ( R ) defined on the Sobolev space W₂²ⁿ ( R ) , where

p (z) = z2n+a2n-1z2n-1+ ... +a1 z+a0

is a polynomial with real coefficients a_j .

If p(t) is nonnegative, then the operator A_p is a definitizable operator. Using M. Krein-H. Langer's spectral theory of definitizable operators B. Curgus and B. Najman proved in [1], that the operator A_p  is similar to a selfadjoint operator if p(t)  is a nonnegative polynomial with at most one real root.

In our work we generalize this result and obtain the following criterion for an operator A_p to be similar to a selfadjoint one:

Theorem 1 The operator A_p = ( sgn x ) p( -i[ d/dx] ) is similar to a selfadjoint operator if and only if the polynomial p(t) is nonnegative.

Our approach is completely different. It is based on the Carleson embedding theorem and the similarity criterion which has been obtained by S. N. Naboko [3] and M. M. Malamud [2].

References:

1. Curgus B., Najman B. Positive differential operator in Krein space L² ( R ) // Oper. Theory Adv. Appl., Birkhauser, Basel.-1996.-87. P.95-104.

2. Malamud M. M. A criterion of the similarity of a closed operator to a selfadjoint operator // Ukrainian Math. J.-1985.-37, no.1.-P.49-56. (Russian)

3. Naboko S. N. On some conditions of similarity to unitary and selfadjoint operators // Func. anal. and its applic.-1984.-18, no.1.-P.16-27. (Russian)

(T)

Date received: January 6, 2000