Let W(𝜃) be a real
valued measurable function on (−∞,∞) which is periodic of period 2π. If
W(𝜃) ∈ L2(0,2J.), the associated Toeplitz operator may be defined as a closed
symmetric linear transformation T = TW in Hilbert space. A generalized resolvent
R(w) for T is constructed, along with the corresponding spectral function
P(t). The theory of positive definite functions is used to exhibit a minimal
dilation E(t) for P(t) and to compute its spectral invariants. The main results
of the paper only require that log+|W(𝜃)|∈ L1(0,2π), although with this
hypothesis a Toeplitz operator cannot be defined in the usual way. However, the
operator-valued analytic function R(w) does exist, and the main results
concern its structure. In certain cases R(w) is the resolvent of a selfadjoint
transformation.
