Vol. 31, No. 2, 1969

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On the theory of unbounded Toeplitz operators

James L. Rovnyak

Vol. 31 (1969), No. 2, 481–496

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.

Mathematical Subject Classification
Primary: 47.30
Received: 31 January 1969
Published: 1 November 1969
James L. Rovnyak