Geodesic
On the construction of the square root for some differential operators
Contemporary Mathematics. Fundamental Directions, Proceedings of the Voronezh Winter Mathematical Krein School — 2024, Tome 70 (2024) no. 4, pp. 636-642

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Using the Balakrishnan–Yosida approach to constructing fractional powers of linear operators in a Banach space by means of strongly continuous semigroups with densely defined generating operators, in this paper, a similar scheme is presented for constructing fractional powers of nondensely defined operators by means of semigroups with a summable singularity. It is found that the newly constructed semigroups also have a singularity at zero, and their sharp estimate is established, related to the order of the singularity of the original semigroup and the fractional power of the constructed operator, in particular, the square root. As an example, the obtained results are applied to semigroups with a singularity given in the paper [3] and in the doctoral dissertation of Yu. T. Silchenko, and a square root is also constructed for a nondensely defined operator.
Keywords: strongly continuous semigroups, semigroups with singularity, generating operators, fractional powers of operators. 
V. A. Kostin; D. V. Kostin; M. N. Silaeva. On the construction of the square root for some differential operators. Contemporary Mathematics. Fundamental Directions, Proceedings of the Voronezh Winter Mathematical Krein School — 2024, Tome 70 (2024) no. 4, pp. 636-642. http://geodesic.mathdoc.fr/item/CMFD_2024_70_4_a9/
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[1] K. Yosida, Functional Analysis, Russian translation, Mir, M., 1967 | MR

[2] S. G. Kreyn, Linear Differential Equations in a Banach space, Nauka, M., 1967 (in Russian)

[3] Yu. T. Sil'chenko, “Solvability of the Cauchy problem for a second-order linear equation with nondensely defined operator coefficients generating semigroups with singularities”, Bull. Higher Edu. Inst. Ser. Math., 1993, no. 11, 40–49 (in Russian) | Zbl

[4] P. E. Sobolevskiy, “On second-order differential equations in a Banach space”, Rep. Acad. Sci. USSR, 146:4 (1962), 774–777 (in Russian) | Zbl

[5] V. V. Uchaykin, Fractional Derivative Methods, Logos, Ul'yanovsk, 2002 (in Russian)