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Symbols and homotopic classification of families of one-dimensional singular operators with piecewise-continuous coefficients. (English. Russian original) Zbl 0685.47031

Sov. Math. 32, No. 12, 23-39 (1988); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1988, No. 12(319), 17-27 (1988).
Let \(\Gamma\) is a simple closed oriented Lyapunov contour, \(L^ n_ p(\Gamma)\) the space n-dimensional vector-functions with \(L_ p(\Gamma)\) elements \((1<p<\infty)\), End \(L^ n_ p(\Gamma)\) the Banach algebra of all linear bounded operators in \(L^ n_ p(\Gamma)\), \({\mathcal M}^ n_ p(\Sigma)\) (\(\Sigma\) \(\subset \Gamma)\) the closed subalgebra of End \(L^ n_ p(\Gamma)\) generated by the operator \(S_{\Gamma}\) \[ (S_{\Gamma}f)(t)=\frac{1}{\pi i}\int_{\Gamma}\frac{f(\tau)d\tau}{\tau -t} \] and operators of multiplication on the piecewise-continuous (n\(\times n)\) matrix-functions on \(\Gamma\). These matrix-function can have a point of discontinuity of first kind only. Let \(\Sigma\) be the set of these points. The authors introduce a new form of Fredholm property for operators from \({\mathcal M}^ n_ p(\Sigma)\) describe the symbol algebras and give a homotopic classification of families of operators from \({\mathcal M}^ n_ p(\Sigma)\).
Reviewer: V.S.Rabinovich

MSC:

47B38 Linear operators on function spaces (general)
47A53 (Semi-) Fredholm operators; index theories
45E05 Integral equations with kernels of Cauchy type
45F15 Systems of singular linear integral equations
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
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