A geometric approach to numerical radius inequalities

In this article we present new numerical radius inequalities for that if A is a sectorial matrix with sectorial index-y E [0, pi sectorial matrices, and their analogues. For example, we show ),2 then IIAA* + A*AII 2(1 + sin2-y) < w(A)2, where w(A) is the numerical radius of A. The significance of this result is the geometric meaning of the obtained inequality. In particular, we will show how this interpolates some well known inequalities in the literature, in a way that depends on the angle of the sector that contains the numerical range of the sectorial matrix A. Many other interpolating inequalities will be presented to show how the ratio HA- w(A) depends mainly on the angle of the sector that contains the numerical range of the matrix A. Our discussion will lead to a new vision about those matrices which contain the origin in the interior of their numerical ranges.(c) 2022 Elsevier Inc. All rights reserved.