In this paper, we give several upper and lower bounds for the numerical radius of 2 ×2 block matrices. Several special cases of our results are given.

In this paper, we present several singular value inequalities for special types of functions of products and sums of matrices. Some of special cases of our results give a generalization of some recent inequalities

In this paper, it is shown, among other inequalities, that if \(A,B\in \mathcal{\mathbb{B(H)}}\), then, for \(p\geq 1,\) we have% \[ 2^{\frac{1}{p}-2}\left\Vert \left\vert A^{\ast }\right\vert ^{2}+\left\vert B\right\vert ^{2}\right\Vert _{p}\leq 2^{\frac{1}{p}-3}\left( \left\Vert A^{\ast }+B\right\Vert _{2p}^{2}+\left\Vert A^{\ast }-B\right\Vert _{2p}^{2}\right) \leq w_{2p}^{2}\left( \left[ \begin{array}{cc} 0 & A \\ B & 0% \end{array}% \right] \right) \] and% \[ w_{2p}^{2}\left( \left[ \begin{array}{cc} 0 & A \\ B & 0% \end{array}% \right] \right) \leq 2^{\frac{1}{p}-1}\left( \left\Vert \left\vert A\right\vert ^{2}\right\Vert _{p}+\left\Vert \left\vert B^{\ast }\right\vert ^{2}\right\Vert _{p}\right) -(2^{\frac{1}{p}-1}-1)\left\vert \left\Vert \left\vert A\right\vert ^{2}\right\Vert _{p}-\left\Vert \left\vert B^{\ast }\right\vert ^{2}\right\Vert _{p}\right\vert . \]

In this article, we derive various lower bounds for the numerical radius of operators that refine the well-known inequality \(w^{2}(A)\geq \frac{1}{4} \left\Vert A^{\ast }A+AA^{\ast }\right\Vert \).

In this article, we introduce some inner product inequalities for operators involving the Cartesian decomposition of the operator. We employ the obtained results to get norm and numerical radius inequalities that extend and improve some earlier inequalities.

In this paper, we define several types of weakly covering spaces concerning coc-open sets, we study some implications between these weakly covering spaces. Also, we give several product theorems under coc-almost open function.