Introduction

The Toeplitz operatorT(a) on the real line ℝ induced by a function a given on ℝ is defined by 1$$T(a)f = {P^ + }(af),$$ where P+ is the analytic projector of the space L _p (ℝ) , 1 < p < ∞, onto the Hardy subspace H _p (∏₊) for the upper complex half-plane ∏₊. The function a is called the symbol of the operator T(a). To study such operators in the Hardy space H _p (∏₊) we will assume from now on that a ∈ L∞(ℝ). In the simplest case the symbol a admits a representation 2$$a(x) = {a^ + }(x){\left( {\frac{{x - i}}{{x + i}}} \right)^\kappa }{a^ - }(x),$$ where 3$${({a^ + })^{ \pm 1}} \in {H^\infty }({\prod _ + }), {({a^ - })^{ \pm 1}} \in {H^\infty }({\prod _ - }), \kappa \in \mathbb{Z},$$ which is known as a canonical factorization. The rational function 4$$r(x) = {\left( {\frac{{x - i}}{{x + i}}} \right)^\kappa }$$ plays a determining role in the description of the main qualitative characteristics of the operator T(a). Specifically, for κ > 0 the operator T(a) is left-invertible and dim coker T(a) = κ for κ < 0 it is right-invertible and dim ker T(a) = −κ and for κ = 0 it is invertible. As a consequence, the index of the operator T(a) is equal to −κ, and is also equal to minus the topological index of the function r, which is defined by $$in{d_\mathbb{R}}r = \frac{1}{{2\pi }}\arg r(x)|_{ - \infty }^\infty .$$