Some numerical radius bounds

In this paper, we first give two new upper bounds for the numerical radius of the product of two Hilbert space operators. The obtained bounds are compared numerically with previously known bounds. After that, the Hilbert–Schmidt numerical radius is studied for the generalized Aluthge transform and a pair of commuting operators. In the end, off-diagonal, tridiagonal, and anti-tridiagonal operator matrices are treated, where many upper bounds that extend some existing results are shown.