The Laguerre Type Operator in a Left Definite Hilbert Space

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This article completes earlier work concerning the Laguerre type differential operator of fourth order, set in a weighted Sobolev space on [0, ∞). It is shown that the operator, which is self-adjoint in L2(0, ∞; e−x) ⊗ R, is also self-adjoint in the new space, whose inner product also involves first and second derivatives. Further, the spectrum remains unchanged, the Laguerre type polynomials are eigenfunctions, and the spectral resolution, associated with the operator, is still an eigenfunction expansion.