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ON THE SPECTRUM OF OPERATORS CONCERNED WITH THE REDUCED SINGULAR CAUCHY INTEGRAL

Authors
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Institute of Mathematics of the National Academy of Sciences of Ukraine
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Keywords
  • Mathematical Analysis

Abstract

Oleg F. Gerus (Zhytomyr, Ukraine) ON THE SPECTRUM OF OPERATORS CONCERNED WITH THE REDUCED SINGULAR CAUCHY INTEGRAL1 Let Γ be a rectifiable closed Jordan curve in the complex plane; Γz,δ := {ζ ∈ Γ : |ζ − z| 6 δ} , δ > 0 ; f : Γ→ C be a continuous function; F[f ](t) = 1 2pii ∫ Γ f(ζ)− f(t) ζ − t dζ := 1 2pii lim δ→0 ∫ Γ\Γt,δ f(ζ)− f(t) ζ − t dζ, t ∈ Γ, be the reduced singular Cauchy integral. The operator F is interpretable in the following shape: F = B+ iC, where B[f ](t) := − 1 2pi ∫ Γ f(ζ)− f(t) |ζ − t|2 ((η − ν)dξ − (ξ − σ)dη), t ∈ Γ, C[f ](t) := − 1 2pi ∫ Γ f(ζ)− f(t) |ζ − t|2 ((ξ − σ)dξ + (η − ν)dη), t ∈ Γ, and ζ := ξ + iη , t := σ + iν . We investigate spectrums of the operators F , B and C . Among others we proved that numbers 0 and −1 are proper numbers of infinite order of the operator F . The point spectrum of the operator B excepting points 0 and −1 is reflected in the point −1 2 . And the point spectrum of the operator C is reflected in the origin. There is a close connection between spectrums of operators B and C , owing to the equalities { C2 −B2 = B, BC+CB = −C. So there exists a one-to-one correspondence between the eigenvalues λ ∈ (−1,−1 2 )∪ (−1 2 , 0) of the operator B and γ = iσ , σ ∈ (−1 2 , 0) ∪ (0, 1 2 ) , of the operator C by the formula σ2 = −λ2 − λ . 1Partially supported by Fundamental Researches State Fund, project 25.1/084. 1

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