  We use a model operator approach and the spectral theorem for self-adjoint
operators in a Hilbert space to derive the basic results of abstract
left-definite theory in a straightforward manner. The theory is amply
illustrated with a variety of concrete examples employing scales of Hilbert
spaces, fractional Sobolev spaces, and domains of (strictly) positive
fractional powers of operators, employing interpolation theory.
  In particular, we explicitly describe the domains of positive powers of the
harmonic oscillator operator in $L^2(\mathbb{R})$ $\big($and hence that of the
Hermite operator in $L^2\big(\mathbb{R}; e^{-x^2}dx)\big)\big)$ in terms of
fractional Sobolev spaces, certain commutation techniques, and positive powers
of (the absolute value of) the operator of multiplication by the independent
variable in $L^2(\mathbb{R})$.
