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<p>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&#xa0;[<CitationRef CitationID="CR2">2</CitationRef>] and in Sil’chenko’s doctoral dissertation, and a square root is also constructed for a nondensely defined operator.</p>

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ON THE CONSTRUCTION OF THE SQUARE ROOT FOR SOME DIFFERENTIAL OPERATORS

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • V. A. Kostin,
  • D. V. Kostin,
  • M. N. Silaeva

摘要

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 [2] and in Sil’chenko’s doctoral dissertation, and a square root is also constructed for a nondensely defined operator.