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Singular Integral Equations with Continuous Coefficients on a Composed Contour

  • Israel Gohberg
  • Nahum Krupnik
Part of the Operator Theory: Advances and Applications book series (OT, volume 206)

Abstract

In this paper the algebra generated by singular integral operators A of the form \( (A\phi )(t) = a(t)\phi (t) + \frac{{b(t)}} {{\pi i}}\int_\Gamma {\frac{{\phi (t)}} {{\tau - t}}d\tau } \left( {t \in \Gamma } \right), \) where Γ is an oriented contour in the complex plane that consists of a finite number of closed and open simple Lyapunov curves, a(t) and b(t) are continuous functions on Γ, is studied. The operators of the form (0.1) will be considered in the space L p (Γ, ϱ) (1 < p < ∞) with weight
$$ \varrho (t) = \prod\limits_{k = 1}^{2N} {\left| {t - \alpha _k } \right|^{\beta _k } } , $$
where α k (k = 1, ..., N) are the starting points and α k (k = N + 1, ..., 2N) are the terminating points of the corresponding open arcs of the contour Γ, and the numbers β k satisfy the conditions –1 < β k < p–1. In what follows we will denote the space L p (Γ, ϱ) by L υ, where the vector υ is defined by υ = (p, β 1, ..., β 2N ).

Keywords

English Translation Matrix Function Maximal Ideal Banach Algebra Singular Integral Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Basel AG 2010

Authors and Affiliations

  • Israel Gohberg
  • Nahum Krupnik

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