The Q-value Field Of The Operator

In this paper, we mainly study the q-numerical range, which is the extension of the classical numerical range. We first give the definition of the q-numerical range and the q-numerical radius, introduce the basic properties of the q-numerical range and some results of the q-numerical range. Then we introduce the norm property and some inequalities of the q-numerical radius. Finally, through the extension of the algebraic properties of W{0}={A ∈B(H),0 ∈ W(A)}, when in the situation of the q-numerical range, we give the algebraic properties of Wq{0}={A ∈B(H),0 ∈ Wq(A)}.According to the contents, this paper is divided into three chapters.In chapter 1, we first give the definition of the numerical range and the numer-ical radius, the definition of the q-numerical range and the q-numerical radius, then introduce some results of Wq(A), including the convexity of Wq(A), the relationship between Wq(A) when q is different, the characterization of Wq(A) and the boundary of Wq(A).In chapter 2, we introduce the norm property and inequalities of rq(A), extend the inequalities of r(A) to rq(A).In chapter 3, let We extend the algebraic properties of W(0) to Wq{0}.