The Legendre operator function

We study the properties of the resolving operator of the Cauchy problem for an abstract Legendre equation with an unbounded linear operator $A$. We establish a relationship between this resolving operator and that of the Cauchy problem for an abstract Euler–Poisson–Darboux equation and prove the coincidence of the sets of operators $A$ for which the Cauchy problems for the Legendre and Euler–Poisson–Darboux equations are uniformly well posed.