Littlejohn and Wellman developed a general abstract left-definite theory for a self-adjoint operator A that is bounded below in a Hilbert space (H, (·,·)). More specifically, they construct a continuum of Hilbert spaces {(Hr, (·,·)r)}r >0 and, for each r > 0, a self-adjoint restriction Ar of A in Hr. The Hilbert space Hr is called the r th left-definite Hilbert space associated with the pair ( H,A) and the operator Ar is called the rth left-definite operator associated with (H,A). We apply this left-definite theory to the self-adjoint Legendre type differential operator generated by the fourth-order formally symmetric Legendre type differential expression ℓyx :=&parl0;&parl0;1-x2&parr0;2y''&parl0;x &parr0;&parr0;''-&parl0;&parl0;8+4A&parl0;1-x2&parr0; &parr0;y'&parl0;x&parr0;&parr0;'+ly&parl0;x&parr0;, where the numbers A and lambda are, respectively, fixed positive and non-negative parameters and where x ∈ (--1, 1).
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