Abstract
The study of the spectral properties of operators generated by differential equations of hyperbolic or parabolic type with Cauchy initial data involve, as a rule, Volterra boundary-value problems that are well posed. But Hadamard’s example shows that the Cauchy problem for the Laplace equation is ill posed. At present, not a single Volterra well-defined restriction or extension for elliptic-type equations is known. Thus, the following question arises: Does there exist a Volterra well-defined restriction of a maximal operator $$\hat L$$ or a Volterra well-defined extension of a minimal operator L 0 generated by the Laplace operator? In the present paper, for a wide class of well-defined restrictions of the maximal operator $$\hat L$$ and of well-defined extensions of the minimal operator L 0 generated by the Laplace operator, we prove a theorem stating that they cannot be Volterra.