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Search Results (3)

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Keywords = strictly accretive operator

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21 pages, 330 KiB  
Article
Schatten Index of the Sectorial Operator via the Real Component of Its Inverse
by Maksim V. Kukushkin
Mathematics 2024, 12(4), 540; https://doi.org/10.3390/math12040540 - 8 Feb 2024
Viewed by 899
Abstract
In this paper, we study spectral properties of non-self-adjoint operators with the discrete spectrum. The main challenge is to represent a complete description of belonging to the Schatten class through the properties of the Hermitian real component. The method of estimating the singular [...] Read more.
In this paper, we study spectral properties of non-self-adjoint operators with the discrete spectrum. The main challenge is to represent a complete description of belonging to the Schatten class through the properties of the Hermitian real component. The method of estimating the singular values is elaborated by virtue of the established asymptotic formulas. The latter fundamental result is advantageous since, of many theoretical statements based upon it, one of them is a concept on the root vectors series expansion, which leads to a wide spectrum of applications in the theory of evolution equations. In this regard, the evolution equations of fractional order with the sectorial operator in the term not containing the time variable are involved. The concrete well-known operators are considered and the advantage of the represented method is convexly shown. Full article
(This article belongs to the Special Issue Direct and Inverse Spectral Problems for Ordinary Differential and Functional-Differential Operators)
27 pages, 403 KiB  
Article
Natural Lacunae Method and Schatten–Von Neumann Classes of the Convergence Exponent
by Maksim V. Kukushkin
Mathematics 2022, 10(13), 2237; https://doi.org/10.3390/math10132237 - 26 Jun 2022
Cited by 8 | Viewed by 1384
Abstract
Our first aim is to clarify the results obtained by Lidskii devoted to the decomposition on the root vector system of the non-selfadjoint operator. We use a technique of the entire function theory and introduce a so-called Schatten–von Neumann class of the convergence [...] Read more.
Our first aim is to clarify the results obtained by Lidskii devoted to the decomposition on the root vector system of the non-selfadjoint operator. We use a technique of the entire function theory and introduce a so-called Schatten–von Neumann class of the convergence exponent. Considering strictly accretive operators satisfying special conditions formulated in terms of the norm, we construct a sequence of contours of the power type that contrasts the results by Lidskii, where a sequence of contours of the exponential type was used. Full article
20 pages, 387 KiB  
Article
Evolution Equations in Hilbert Spaces via the Lacunae Method
by Maksim V. Kukushkin
Fractal Fract. 2022, 6(5), 229; https://doi.org/10.3390/fractalfract6050229 - 20 Apr 2022
Cited by 7 | Viewed by 2113
Abstract
In this paper, we consider evolution equations in the abstract Hilbert space under the special conditions imposed on the operator at the right-hand side of the equation. We establish the method that allows us to formulate the existence and uniqueness theorem and find [...] Read more.
In this paper, we consider evolution equations in the abstract Hilbert space under the special conditions imposed on the operator at the right-hand side of the equation. We establish the method that allows us to formulate the existence and uniqueness theorem and find a solution in the form of a series on the root vectors of the right-hand side. We consider fractional differential equations of various kinds as an application. Such operators as the Riemann-Liouville fractional differential operator, the Riesz potential, the difference operator have been involved. Full article
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