The fourth-order Legendre type differential expression Mkis studied in the spaces L2(-1,1) and L2μ[-1,1], where μ is the orthogonalizing weight for the Legendre type polynomials. In L2(-1,1), Mkis 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], Mkgenerates a self-adjoint operator which is self-adjoint on the maximal domain of Mkin 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.
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