ON SOME PROPERTIES OF THE LEGENDRE TYPE DIFFERENTIAL EXPRESSION

Abstract

The fourth-order Legendre type differential expression M_k is studied in the spaces L²(-1,1) and L^{2 μ}[-1,1], where μ is the orthogonalizing weight for the Legendre type polynomials. In L²(-1,1), M^k is found to be limit-8 at both endpoints. Consequently, limiting boundary values and self-adjoint restrictions of the maximal operator are determined. The spectra of these operators are shown to be discrete. In L²[-1,1], M_k generates a self-adjoint operator which is self-adjoint on the maximal domain of M_k in L²(-1,1); its spectrum is also discrete. The Legendre type polynomials form a complete orthogonal set in this space and also in the associated left-definite Sobolev space H.