Equations with Operators Which Act in a Single Space

Abstract

Let A be an operator which acts in the space E, has a domain dense in E, and is Noetherian. The operators A², A³ ,... enjoy these same properties, and this allows us to introduce a new important characteristic of equation (A). Namely, let W_∞ (A) be the set of all elements x such that A^kx = θ for some k > 0. This set is a linear manifold: if A^k1x₁ = θ and A^k2x₂ = θ, then

$$ {A^{k}}\left( {{x_{1}} + {x_{2}}} \right) = \theta {\kern 1pt} for{\kern 1pt} k = \max \left( {{k_{1}},{k_{2}}} \right) $$

.