Abstract Evolution Equations with an Operator Function in the Second Term
Abstract
:1. Introduction
2. Preliminaries
- C—a real positive constant, we assume that a value of C can be different within a formula.
- —the set of linear bounded operators on
- —the domain of definition of the operator
- —the range of the operator
- —the kernel of the operator
- —the resolvent set of the operator
- —the eigenvalues of the operator
- —the eigenvalues of the operator
- —the singular numbers (s-numbers) of the compact operator where
- —the function equals to the number of the elements of the sequence within the circle
- —the counting function a function corresponding to the sequence where T is a compact operator.
- —the Schatten-von Neumann class.
- —the set of compact operators.
- —the Schatten-von Neumann class of the convergence exponent, defined as follows
- —the numerical range of the operator defined as follows
- — the interior of the set
- —the set of boundary points of the set
- —a closed sector with the vertex and the semi-angle of the sector defined as follows If we want to stress the correspondence between and then we will write
- — the maximum absolute value of the function f on the boundary of the circle with the radios r and the center at point zero.
- — the minimum absolute value of the function f on the boundary of the circle with the radios r and the center at point zero.
- Convergence in the Abel-Lidsky sense
3. Main Results
3.1. Existence and Uniqueness Theorems
3.2. Concrete Operators
4. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Kukushkin, M.V. Natural lacunae method and Schatten-von Neumann classes of the convergence exponent. Mathematics 2022, 10, 2237. [Google Scholar] [CrossRef]
- Kukushkin, M.V. Evolution Equations in Hilbert Spaces via the Lacunae Method. Fractal Fract. 2022, 6, 229. [Google Scholar] [CrossRef]
- Agranovich, M.S. On series with respect to root vectors of operators associated with forms having symmetric principal part. Funct. Anal. Its Appl. 1994, 28, 151–167. [Google Scholar] [CrossRef]
- Gohberg, I.C.; Krein, M.G. Introduction to the Theory of Linear Non-Selfadjoint Operators in a Hilbert Space; Nauka, Fizmatlit: Moscow, Russia, 1965. [Google Scholar]
- Lidskii, V.B. Summability of series in terms of the principal vectors of non-selfadjoint operators. Tr. Mosk. Mat. Obs. 1962, 11, 3–35. [Google Scholar]
- Kukushkin, M.V. On One Method of Studying Spectral Properties of Non-selfadjoint Operators. In Abstract and Applied Analysis; Hindawi: London, UK, 2020. [Google Scholar] [CrossRef]
- Kukushkin, M.V. Asymptotics of eigenvalues for differential operators of fractional order. Fract. Calc. Appl. Anal. 2019, 22, 658–681. Available online: https://www.degruyter.com/view/j/fca (accessed on 1 January 2022). [CrossRef]
- Katsnelson, V.E. Conditions under which systems of eigenvectors of some classes of operators form a basis. Funct. Anal. Appl. 1967, 1, 122–132. [Google Scholar] [CrossRef]
- Krein, M.G. Criteria for completeness of the system of root vectors of a dissipative operator. Amer. Math. Soc. Transl. Ser. 1963, 26, 221–229. [Google Scholar]
- Markus, A.S.; Matsaev, V.I. Operators generated by sesquilinear forms and their spectral asymptotics. Mat. Issled 1981, 61, 86–103. [Google Scholar]
- Markus, A.S. Expansion in root vectors of a slightly perturbed selfadjoint operator. Soviet Math. Dokl. 1962, 3, 104–108. [Google Scholar]
- Motovilov, A.K.; Shkalikov, A.A. Preserving of the unconditional basis property under non-self-adjoint perturbations of self-adjoint operators. Funktsional. Anal. Prilozhen. 2019, 53, 45–60. [Google Scholar]
- Shkalikov, A.A. Perturbations of selfadjoint and normal operators with a discrete spectrum. Russ. Math. Surv. 2016, 71, 113–174. [Google Scholar] [CrossRef]
- Kukushkin, M.V. Abstract fractional calculus for m-accretive operators. Int. J. Appl. Math. 2021, 34. [Google Scholar] [CrossRef]
- Rozenblyum, G.V.; Solomyak, M.Z.; Shubin, M.A. Spectral theory of differential operators. In Results of Science and Technology; Series Modern Problems of Mathematics Fundamental Directions; Springer: Berlin/Heidelberg, Germany, 1989; Volume 64, pp. 5–242. [Google Scholar]
- Kipriyanov, I.A. The operator of fractional differentiation and powers of the elliptic operators. Proc. Acad. Sci. USSR 1960, 131, 238–241. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives: Theory and Applications; Gordon and Breach Science Publishers: Philadelphia, PA, USA, 1993. [Google Scholar]
- Andronova, O.A.; Voytitsky, V.I. On spectral properties of one boundary value problem with a surface energy dissipation. UFA Math. J. 2017, 9, 3–16. [Google Scholar] [CrossRef]
- Mamchuev, M.O. Boundary value problem for the time-fractional telegraph equation with Caputo derivatives Mathematical Modelling of Natural Phenomena. Spec. Funct. Anal. PDEs 2017, 12, 82–94. [Google Scholar] [CrossRef] [Green Version]
- Mamchuev, M.O. Solutions of the main boundary value problems for the time-fractional telegraph equation by the Green function method. Fract. Calc. Appl. Anal. 2017, 20, 190–211. [Google Scholar] [CrossRef]
- Pskhu, A.V. The fundamental solution of a diffusion-wave equation of fractional order. Izv. Math. 2009, 73, 351–392. [Google Scholar] [CrossRef]
- Moroz, L.; Maslovskaya, A.G. Hybrid stochastic fractal-based approach to modeling the switching kinetics of ferroelectrics in the injection mode. Math. Model. Comput. Simul. 2020, 12, 348–356. [Google Scholar] [CrossRef]
- Levin, B.J. Distribution of Zeros of Entire Functions. In Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, USA, 1964. [Google Scholar]
- Kato, T. Perturbation Theory for Linear Operators; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1980. [Google Scholar]
- Riesz, F.; Nagy, B.S. Functional Analysis; Ungar: New York, NY, USA, 1955. [Google Scholar]
- Hardy, G.H. Divergent Series; Oxford University Press: London, UK, 1949. [Google Scholar]
- Krasnoselskii, M.A.; Zabreiko, P.P.; Pustylnik, E.I.; Sobolevskii, P.E. Integral Operators in the Spaces of Summable Functions; Science Fizmatlit: Moscow, Russia, 1966. [Google Scholar]
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Kukushkin, M.V. Abstract Evolution Equations with an Operator Function in the Second Term. Axioms 2022, 11, 434. https://doi.org/10.3390/axioms11090434
Kukushkin MV. Abstract Evolution Equations with an Operator Function in the Second Term. Axioms. 2022; 11(9):434. https://doi.org/10.3390/axioms11090434
Chicago/Turabian StyleKukushkin, Maksim V. 2022. "Abstract Evolution Equations with an Operator Function in the Second Term" Axioms 11, no. 9: 434. https://doi.org/10.3390/axioms11090434
APA StyleKukushkin, M. V. (2022). Abstract Evolution Equations with an Operator Function in the Second Term. Axioms, 11(9), 434. https://doi.org/10.3390/axioms11090434