Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Improved inequalities related to the $A$-numerical radius for
  commutators of operators   Let $A$ be a positive bounded linear operator on a complex Hilbert space
$\mathcal{H}$ and $\mathcal{B}_{A}(\mathcal{H})$ be the subspace of all
operators which admit $A$-adjoints operators. In this paper, we establish some
inequalities involving the commutator and the anticommutator of operators in
semi-Hilbert spaces, i.e. spaces generated by positive semidefinite
sesquilinear forms. Mainly, among other inequalities, we prove that for $T,
S\in\mathcal{B}_{A}(\mathcal{H})$ we have \begin{align*} \omega_A(TS \pm ST)
\leq 2\sqrt{2}\min\Big\{f_A(T,S), f_A(S,T) \Big\}, \end{align*} where
$$f_A(X,Y)=\|Y\|_A\sqrt{\omega_A^2(X)-\frac{\left|\,\left\|\frac{X+X^{\sharp_A}}{2}\right\|_A^2-\left\|\frac{X-X^{\sharp_A}}{2i}\right\|_A^2\right|}{2}}.$$
Here $\omega_A(\cdot)$ and $\|\cdot\|_A$ are the $A$-numerical radius and the
$A$-operator seminorm of semi-Hilbert space operators, respectively and
$X^{\sharp_A}$ denotes a distinguished $A$-adjoint operator of $X$.

12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
13
The categories of this paper are: quant-ph
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
27
The categories of this paper are: comp-gas
32
The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
19
The categories of this paper are: eess