On the construction of the square root for some differential operators

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.