Let \(\mathbb{B}(X)\) be the algebra of all bounded linear operators on a Hilbert space X. Consider an operator polynomial \(P(\lambda)=A_{m}\lambda^{m}+A_{m-1}\lambda^{m-1}+\cdots+A_{0},\) where \(A_{i}\in\mathbb{B}(X),i=0,1,\cdots,m\) . The numerical range of P(λ) is defined as \(W(P(\lambda))=\{\lambda\in\mathbb{C}:(P(\lambda)x,x)=0\;\text{for}\;\text{some}\;x\ne0\}.\) The main goal of this paper is to respond to an open problem proposed by professor Li, and determine general conditions on connectivity, convexity and spectral inclusion property of W(P(λ)). They also consider the relationship between operator polynomial numerical range and block numerical range.
