Lower and upper bounds for the $p$-(A-M)-norm of two operators in Hilbert spaces with applications

For $\nu \in [0,1]$, $p\ge 1$ and $A,B\in \mathcal{B}\left(H\right)$, we define the $p$-arithmetic-mean (A-M)-norm for the pair of operators $\left(A,B\right)$ by

$${\Vert \left(A,B\right)\Vert }_{p,\nu }:=\underset{\Vert x\Vert =1}{sup}{\left(\left(1-\nu \right){\Vert Ax\Vert }^p+\nu {\Vert Bx\Vert }^p\right)}^{1/p}.$$

In this paper, we obtain several lower and upper bounds for this norm. Some inequalities for the numerical radius of the off-diagonal operator matrix are given. In the case when $\left(A,B\right)=\left(T,T^{\ast }\right)$ and $\left(A,B\right)=\left(\mathrm{R}\mathrm{e}T,\mathrm{I}\mathrm{m}T\right)$, where $\mathrm{R}\mathrm{e}T:=\frac{T+T^{\ast }}{2}$ is the real part of $T$ and $\mathrm{I}\mathrm{m}T:=\frac{T-T^{\ast }}{2\text{i}}$ is the imaginary part of $T$, respectively, some inequalities for one operator are also provided.