Nahum Krupnik

The Spectrum of Singular Integral Operators in L p Spaces

First, we shall consider the simplest class of one-dimensional singular integral operators — the class of discrete Wiener-Hopf operators.

One-dimensional Singular Integral Operators with Shift

Let Г be a closed or open oriented Lyapunov arc and ω(t) be a bijective mapping of Г onto itself. An operator of the form \( A = a(t)I + b(t)S + (c(t) + d(t)S)W \) is usually called a one-dimensional singular integral operator with shift ω(t). Here a(t), b(t), c(t), and d(t) are bounded measurable functions on Г, S is the operator of singular integration along Г given by

On an Algebra Generated by the Toeplitz Matrices in the Spaces h p

(S_{\Gamma \phi } )(t) = \frac{1} {{\pi i}}\int_\Gamma {\frac{{\phi (\tau )}} {{\tau - t}}d\tau } (t \in \Gamma ),

On Singular Integral Equations with Unbounded Coefficients

and W is the shift operator defined by

Algebras of Singular Integral Operators with Shift

(W_\phi )(t) = \phi (\omega (t)).