On the $p$-arithmetic-mean norm of operator pairs and its applications

The main purpose of this paper is to introduce and study the so-called $p$-arithmetic-mean norm for pairs of bounded linear operators on a complex Hilbert space $H$. Specifically, for a pair $(A,B)$ of bounded linear operators on $H$, with $\nu \in [0,1]$ and $p>0$, we define the following:

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

We establish, among other results, that for $\nu \in (0,1]$ and $p\in (0,1]$,

$$\Vert (A,B){\Vert }_{2p,\nu }^{2p}\le R\Vert A-B{\Vert }^{2p}+min\{\Vert A{\Vert }^{2(1-\nu )p}\Vert B{\Vert }^{2\nu p},\Vert (1-\nu )|A{|}^2+\nu |B{|}^2{\Vert }^p\}.$$

Applications are given to off-diagonal operator matrices, and to the particular cases $(A,B)=(T,T^{\ast })$ and $(A,B)=(\mathrm{Re}(T),\mathrm{Im}(T))$, where $T$ is a bounded linear operator on $H$, $T^{\ast }$ denotes its adjoint, $\mathrm{Re}T=\frac{1}{2}(T+T^{\ast })$ is its real part, and $\mathrm{Im}T=\frac{1}{2i}(T-T^{\ast })$ is its imaginary part.