315 research outputs found
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 is an accretive-dissipative
bounded linear operator on a Hilbert space, then , where and 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 and many other related forms, as a
new discussion in this field; where and are the
real and imaginary parts of the operator . 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
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