In this article, we present new inequalities for the numerical radius of sum two Hilbert space operators. These will enable us to obtain many generalizations and refinements some well known inequalities, including multiplicative behavior norm bounds. Among other applications, it is shown that if T accretive-dissipative, then 1/?2 ||T|| ? ?(T), where ?(?) ||?||denote usual operator norm, respect...

Let X be a Banach space with the Radon-Nikodỳm property. Then, the following are equivalent. (i) X has numerical index 1. (ii) |x∗∗(x∗)| = 1 for all x∗ ∈ ex(BX∗ ) and x∗∗ ∈ ex(BX∗∗ ). (iii) X is an almost-CL-space. (iv) There are a compact Hausdorff space K and a linear isometry J : X → C(K) such that |x∗∗(J∗δs)| = 1 for all s ∈ K and x∗∗ ∈ ex(BX∗∗ ). If X is a real space, the above conditions ...

Let B(H) be the algebra of all bounded linear operators on a Hilbert space H and let N(⋅) norm B(H). For every 0≤ν≤1, we introduce w(N,ν)(A) as an extension classical numerical radius byw(N,ν)(A):=supθ∈R⁡N(νeiθA+(1−ν)e−iθA⁎) investigate basic properties this notion prove inequalities involving it. In particular, when is Hilbert–Schmidt ‖⋅‖2, present several weighted for operator matrices. Furth...

Utilizing the linking algebra of a Hilbert $C^*$-module $\big(\mathscr{V}, {\|\!\cdot\!\|}_{_{\mathscr{V}}}\big)$, we introduce $\Omega(x)$ as definition numerical radius for an element $x\in\mathscr{V}$ and then show that $\Omega(\cdot)$ is norm on $\mathscr{V}$ such $\frac{1}{2}{\|x\|}_{_{\mathscr{V}}} \leq \Omega(x) {\|x\|}_{_{\mathscr{V}}}$. In addition, obtain equivalent condition $\Omega(...