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...

We show that a bounded linear operator A ∈ B(H) is a multiple of a unitary operator if and only if AZ and ZA always have the same numerical radius or the same numerical range for all (rank one) Z ∈ B(H). More generally, for any bounded linear operators A,B ∈ B(H), we show that AZ and ZB always have the same numerical radius (resp., the same numerical range) for all (rank one) Z ∈ B(H) if and on...

Abstract. The numerical range of a matrix is studied geometrically via the cone of positive semidefinite matrices (or semidefinite cone for short). In particular, it is shown that the feasible set of a two-dimensional linear matrix inequality (LMI), an affine section of the semidefinite cone, is always dual to the numerical range of a matrix, which is therefore an affine projection of the semid...

We show that, for any 2n+ 2 distinct points a1, a′ 1, a2, a′ 2, . . . , an+1, a′ n+1 (in this order) on the unit circle, there is an n-by-n matrix A, unique up to unitary equivalence, which has norm one and satisfies the conditions that it has all its eigenvalues in the open unit disc, In − A∗A has rank one and its numerical range is circumscribed by the two (n+ 1)-gons a1a2 · · · an+1 and a′ 1...