Numerical radius inequalities via block matrices

In this paper, we prove new numerical radius bounds that generalize some well-known results in the literature. For example, we prove that if A, B, X, Y are bounded linear operators on a complex separable Hilbert space H such that A and B are positive, then \(\begin{aligned} w(AX+YB)\le \sqrt{||~A+B~||~||~X^*AX+YBY^*~||}. \end{aligned}\) $$\begin{array}{c}\hfill w(AX+YB)\le \sqrt{||A+B||||X^{\ast }AX+YB{Y}^{\ast }||}.\end{array}$$ This inequality generalizes a celebrated inequality proved by Kittaneh which states that: \(\begin{aligned} w^2(A)\le \frac{1}{2} ||~A^*A+AA^*~||. \end{aligned}\) $$\begin{array}{c}\hfill w^2(A)\le \frac{1}{2}||A^{\ast }A+A{A}^{\ast }||.\end{array}$$ .