On the well-posed solvability of fractional operator equations by the Maslov–Heaviside method

In the paper, for a fractional operator polynomial $P_n(A^\alpha)$, $\alpha\in(0,1)$, with a weakly positive operator $A$ acting on a Banach space, the problem of its bounded invertibility is posed and solved, which is equivalent to the correct Hadamard solvability of the corresponding operator equation. The problem is solved by the Maslov–Heaviside method, which was previously used by the authors in the case of integer powers of the operator $A$. This enables us to obtain an integral representation of the inverse operator of $P_n(A^\alpha)$ using the strongly continuous semigroup $U(t_1-A)$ with the generator $-A$ and to indicate a correctness estimate for this operator; this estimate connects the type of the semigroup with the roots of the scalar polynomial $P_n(x)$ (that was called by V. P. Maslov the symbol of the operator polynomial $P_n(A)$).
Using examples, we show the naturalness of applying the Maslov–Heaviside method to the study of the correct solvability of problems for differential equations.