On the weakly semi-Fredholm structure of a class of singular Hilbert type operator equations

A. K. Prykarpatski; V. M. Dilnyi

doi:10.30970/ms.66.1.80-86

A. K. Prykarpatski Department of Computer Science and Mathematics, Cracow University of Technology Krakow, Poland; Lviv Polytechnic National University Lviv, Ukraine; Ivan Franko State Pedagogical University of Drohobych Lviv region, Ukraine https://orcid.org/0000-0001-5124-5890
V. M. Dilnyi Lviv Polytechnic National University Lviv, Ukraine

DOI: https://doi.org/10.30970/ms.66.1.80-86

Keywords: weakly semi-Fredholm operator, compact operator, singularity, Hilbert transform, operator kernel, Sokhotsky-Plemelj formulae, Riesz-Thorin convexity theorem, Zygmund-Calderon inequality, N. Kolmogorov- F. Riesz compactness theorem

Abstract

We studied the solution space of a linear singular Hilbert type operator equation $T\left( a(z)f(z)\right) +b(z)f(z)=0 $ in the Banach space $L_{q}(\mathbb{C},d\sigma ),q>p>1,$ specified by coefficients $a,b\in L_{r}(\mathbb{C},d\sigma )$ and the operator action $\displaystyle Tf(z)=\frac{1}{\pi } v.p.\int_{\mathbb{C}}\frac{f(\xi )-f(z)}{\left( z-\xi \right) ^{2}}d\sigma (\xi ,\bar{\xi}),$ $f\in L_{p}(\mathbb{C},d\sigma ),$ $r=pq/(q-p).$ Based on classical functional analysis A.N. Kolmogorov and F. Riesz compactness theorems, we formulated conditions under which the the singular operator equation under regard possesses the weakly semi-Fredholm structure.

Author Biographies

A. K. Prykarpatski, Department of Computer Science and Mathematics, Cracow University of Technology Krakow, Poland; Lviv Polytechnic National University Lviv, Ukraine; Ivan Franko State Pedagogical University of Drohobych Lviv region, Ukraine

Department of Computer Science and Mathematics, Cracow University of Technology Krakow, Poland; Lviv Polytechnic National University Lviv, Ukraine; Ivan Franko State Pedagogical University of Drohobych Lviv region, Ukraine

V. M. Dilnyi, Lviv Polytechnic National University Lviv, Ukraine

Lviv Polytechnic National University Lviv, Ukraine

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