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Spectral and algebraic analysis of the fractal Volterra operator on Ck(F)

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  • Uc, Mehmet

Abstract

In this study, the fractal Volterra operator, which is the adapted form of the classical Volterra integral operator to fractal analysis, is defined and the basic structural properties of this new operator are investigated. The operator is defined using the concept of local fractal integration and is proven to be linear, bounded, and quasinilpotent. In order to ensure the validity of these analyses, the fractal Duhamel- type convolution product is introduced on the function space Ck(F) defined in the framework of fractal calculus and it is shown that Ck(F) is a unital, commutative Banach algebra under this structure. In addition, the spectrum of the fractal Volterra operator is determined, and its maximal ideal space is characterized by its singular evaluation function. A solvability criterion for convolution equations is also derived within the new ⋆-Banach algebra setting. Moreover, the classical Weierstrass approximation theorem is generalized in fractal form to the space Ck(F) and it is proven that fractal polynomials are dense in this space. The results obtained provide powerful algebraic and operator theoretic tools that can be used in the analysis of fractal differential equations. In this context, the study presents an original contribution that deepens the interaction of fractal calculus with functional analysis and Banach algebras.

Suggested Citation

  • Uc, Mehmet, 2025. "Spectral and algebraic analysis of the fractal Volterra operator on Ck(F)," Chaos, Solitons & Fractals, Elsevier, vol. 200(P3).
    • Handle: RePEc:eee:chsofr:v:200:y:2025:i:p3:s0960077925010744
    DOI: 10.1016/j.chaos.2025.117061
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    References listed on IDEAS

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    1. Alireza Khalili Golmankhaneh & Renat Timergalievich Sibatov, 2021. "Fractal Stochastic Processes on Thin Cantor-Like Sets," Mathematics, MDPI, vol. 9(6), pages 1-13, March.
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