Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improved inequalities for the numerical radius via Cartesian
  decomposition   We develop various lower bounds for the numerical radius $w(A)$ of a bounded
linear operator $A$ defined on a complex Hilbert space, which improve the
existing inequality $w^2(A)\geq \frac{1}{4}\|A^*A+AA^*\|$. In particular, for
$r\geq 1$, we show that \begin{eqnarray*}\frac{1}{4}\|A^*A+AA^*\|
  \leq\frac{1}{2} \left(
\frac{1}{2}\|\Re(A)+\Im(A)\|^{2r}+\frac{1}{2}\|\Re(A)-\Im(A)\|^{2r}\right)^{\frac{1}{r}}
  \leq w^{2}(A),\end{eqnarray*} where $\Re(A)$ and $\Im(A)$ are the real and
imaginary parts of $A$, respectively. Furthermore, we obtain upper bounds for
$w^2(A)$ refining the well-known upper bound $w^2(A)\leq \frac{1}{2}
\left(w(A^2)+\|A\|^2\right)$. Separate complete characterizations for
$w(A)=\frac{\|A\|}{2}$ and $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ are also
given.

16
The categories of this paper are: stat
29
The categories of this paper are: funct-an
34
The categories of this paper are: chem-ph
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
28
The categories of this paper are: alg-geom
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg