For an $$n\times n$$ complex matrix C, the C-numerical range of a bounded linear operator T acting on Hilbert space dimension at least n is set numbers $$\textrm{tr}\,(CX\,^*\,TX)$$ , where X partial isometry satisfying $$X^*X = I_n$$ . It shown that $$\begin{aligned} \textbf{cl}(W_C(T)) \cap \{\textbf{cl}(W_C(U)): U \hbox { unitary dilation } T\} \end{aligned}$$ for any contraction if and only...

In 2020, Cameron et al. introduced the restricted numerical range of a digraph (directed graph) as tool for characterizing digraphs and studying their algebraic connectivity. Notably, with degenerate polygon (that is, point or line segment) were completely described. this article, we extend those results to include whose is non-degenerate convex polygon. general, refer polygonal. We provide com...

Denote by W (A) the numerical range of a bounded linear operator A, and [A, B] = AB −BA the Lie product of two operators A and B. Let H, K be complex Hilbert spaces of dimension ≥ 2 and Φ : B(H) → B(K) be a map whose range contains all operators of rank ≤ 1. It is shown that Φ satisfies that W ([Φ(A), Φ(B)]) = W ([A, B]) for any A, B ∈ B(H) if and only if dim H = dim K, there exist ε ∈ {1,−1}, ...

For each norm v on <en, we define a numerical range Z., which is symmetric in the sense that Z. =Z"D, where vD is the dual norm. We prove that, for aE <enn, Z.(a) contains the classical field of values V(a). In the special case that v is the lcnorm, Z.(a) is contained in a set G(a) of Gershgorin type defined by C. R. Johnson. When a is in the complex linear span of both the Hennitians and the v...

We apply a method developed by one of the authors, see [1], to localize the numerical range of quasi-sectorial contractions semigroups. Our main theorem establishes a relation between the numerical range of quasi-sectorial contraction semigroups {exp(−tS)}t≥0, and the maximal sectorial generators S. We also give a new prove of the rate O(1/n) for the operator-norm Euler formula approximation: e...

The numerical range W (A) of a bounded linear operator A on a Hilbert space is the collection of complex numbers of the form (Av, v) with v ranging over the unit vectors in the Hilbert space. In terms of the location of W (A), inclusion regions are obtained for W (Ak) for positive integers k, and also for negative integers k if A−1 exists. Related inequalities on the numerical radius w(A) = sup...

Let Mn be the semigroup of n× n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in Mn including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λk(A) is the rank-k numerical range and rk(A) is the rank-k numerical radius of A ∈ Mn. Multiplicative maps φ : S → Mn satisfying rk(φ(A)...