General properties of Toeplitz operators

Abstract

Let 1 <p < ∞, let Γ be a Carleson Jordan curve, and let w be a weight in A _P (Γ). We know that then the operator S is bounded on L ^P(Γ, ω). It follows easily that S² = I, and hence P:= (I + S)/2 is a bounded projection on L ^P(Γ, ω). The image of P, i.e. the space L ₊ ^p(Γ, ω):= PL ^P(Γ, ω), is therefore a closed subspace of L ^P(Γ, ω), which is called the pth Hardy space of Γ and ω. If a ∈ L ^∞(Γ), then the operator of multiplication by a is obviously bounded on L ^P(Γ, ω). The compression of this operator to L ^p ₊ (Γ, ω) is referred to as the Toeplitz operator on L ^p ₊ (Γ, ω) with the symbol a and is denoted by T(a). In other words, T(a) is the bounded operator which sends g ∈ L ^p ₊ (Γ, ω) to P(ag) ∈ L ^p ₊ (Γ, ω). A central problem in the spectral theory of singular integral operators is the determination of the essential spectrum of Toeplitz operators with piecewise continuous symbols. This problem will be completely solved in Chapter 7.