A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

We show that any self-adjoint operator A (bounded or unbounded) in a Hilbert space H=(V,(·,·)) that is bounded below generates a continuum of Hilbert spaces {Hr}r>0 and a continuum of self-adjoint operators {Ar}r>0. For reasons originating in the theory of differential operators, we call each Hr the rth left-definite space and each Ar the rth left-definite operator associated with (H,A). Each space Hr can be seen as the closure of the domain D(Ar) of the self-adjoint operator Ar in the topology generated from the inner product (Arx,y) (x,y∈D(Ar)). Furthermore, each Ar is a unique self-adjoint restriction of A in Hr. We show that the spectrum of each Ar agrees with the spectrum of A and the domain of each Ar is characterized in terms of another left-definite space. The Hilbert space spectral theorem plays a fundamental role in these constructions. We apply these results to two examples, including the classical Laguerre differential expression l[·] in which we explicitly find the left-definite spaces and left-definite operators associated with A, the self-adjoint operator generated by l[·] in L2((0,∞);tαe−t) having the Laguerre polynomials as eigenfunctions.