On the Δ-Finite Operators and their Applications

Let $\mathcal{H}$ be a separable infinite dimensional complex Hilbert space, and $\mathcal{L}(\mathcal{H})$ denotes the algebra of all bounded linear operators on $\mathcal{H}$. A new class of operators is introduced, called $\Delta \mathcal{F}(\mathcal{H})$-finite operators and denoted by $\Delta \mathcal{F}(\mathcal{H})$, defined by: \begin{equation*}\Delta \mathcal{F}(\mathcal{H})=\{A \in \mathcal{L}(\mathcal{H}):\|A X A-X-I\| \geq 1,(\forall X \in \mathcal{L}(\mathcal{H}))\}\end{equation*} Basic properties of this class are given and some examples are presented. We prove that a spectraloid operator or class $\mathcal{Y}$ under certain scalar perturbation is Δ-finite operator. We also give an engineering application.