Spectral analysis of powers of the operator \((Vf)(x)=q(x)\int_0^xw(t)f(t)\,dt\) (Q1886110)

The author studies on the spectral analysis of powers of the operators in the form \(V_{q,w}(f(x))= q(x) \int_{0}^{x} w(t)f(t)dt\) . A subspace \(E\) of the Banach space \(X\) is called a cyclic subspace for the operator \(T : X \rightarrow X\) if \(\text{span} \{T^{n}E: n \geq 0 \}=X.\) Let \(\text{Cyc}(T)\) be the set of all cyclic subspaces of an operator \(T\). An operator \(T\) is said to be cyclic in \(X\) if \(1=\inf_{E}\{\dim E: E \in \text{Cyc}(T) \}\). Among other results, the author gives equivalent conditions on the powers of the operator \(V_{q,w}\) to be cyclic and unicellular.