\(\varepsilon \)-Numerical Range of Operator Polynomial

Relying on some ideas of numerical ranges, we introduce, for operators polynomials on a complex Hilbert space, a new notion called the \(\varepsilon \) $\epsilon$ -numerical range of operators polynomials. Some geometrical and topological properties of these \(\varepsilon \) $\epsilon$ -numerical range are proved. Thereby, we achieve a new characterization of \(\varepsilon \) $\epsilon$ -numerical range of operators polynomials on an arbitrary Hilbert space. All our results are new even for \(n\times n\) $n\times n$ -matrix polynomials.