On the Euclidean operator radius and norm

In this paper, we show several bounds for the numerical radius of a Hilbert space operator in terms of the Euclidean operator norm. The obtained forms will enable us to find interesting refinements of celebrated results in the literature. Then, the $f$-operator radius, recently defined as a generalization of the Euclidean operator radius, will be studied. Many upper bounds will be found and matched with existing results that treat the numerical radius. Special cases of this discussion will lead to some refinements and generalizations of some well-established results in the field. Further, numerical examples are given to support our findings, and a simple optimization application will be presented

Numerical Radius Bounds via the Euclidean Operator Radius and Norm

In this paper, we begin by showing a new generalization of the celebrated Cauchy-Schwarz inequality for the inner product. Then, this generalization is used to present some bounds for the Euclidean operator radius and the Euclidean operator norm. These bounds will be used then to obtain some bounds for the numerical radius in a way that extends many well-known results in many cases. The obtained results will be compared with the existing literature through numerical examples and rigorous approaches, whoever is applicable. In this context, more than 15 numerical examples will be given to support the advantage of our findings. Among many consequences, will show that if $T$ is an accretive-dissipative bounded linear operator on a Hilbert space, then ${{\left\| \left( \Re T,\Im T \right) \right\|}_{e}}=\omega \left( T \right)$, where $\omega(\cdot), \|(\cdot,\cdot)\|_e, \Re T$ and $\Im T$ denote, respectively, the numerical radius, the Euclidean norm, the real part and the imaginary part

New Orders Among Hilbert Space Operators

This article introduces several new relations among related Hilbert space operators. In particular, we prove some L\"{o}wener partial orderings among $T, |T|, \mathcal{R}T, \mathcal{I}T, |T|+|T^*|$ and many other related forms, as a new discussion in this field; where $\mathcal{R}T$ and $\mathcal{I}T$ are the real and imaginary parts of the operator $T$. Our approach will be based on proving the positivity of some new matrix operators, where several new forms for positive matrix operators will be presented as a key tool in obtaining the other ordering results. As an application, we present some results treating numerical radius inequalities in a way that extends some known results in this direction, in addition to some results about the singular values