Linear Differential Operators Naimark.pdf
Kapil Grunewald
The original form of symmetric boundary conditions for Lagrange symmetric (formally self-adjoint) ordinary linear differential expressions, was obtained by I.M. Glazman in his seminal paper of 1950. This result described all self-adjoint differential operators, in the underlying Hilbert function space, generated by real even-order differential expressions.
In this paper, a formally J-symmetric, linear differential expression of 2nth order, with complex-valued coefficients, is considered. A number of results concerning the location of the essential spectrum of associated operators are obtained. These are extensions of earlier work dealing with complex Strum-Liouville operators, and include results which, in the real case, are due to Birman, Glazman and others. They lead to criteria, for the non-emptiness of the regularity field, of the corresponding minimal operator-a condition which is needed in the theory of J-selfadjoint extensions.
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