Suppose $A$ and $B$ are two bounded linear operators on $\ell^2(\mathbb{N})$ such that with respect to the standard orthonormal basis, the matrices $A=(a_{mn})_{m,n}$ and $B=(b_{mn})_{m,n}$ consist of non-negative entries. It is not difficult to show that for any $\lambda\in [0,1]$, the matrix $(a_{mn}^{\lambda}b_{mn}^{1-\lambda})_{m,n}$ gives rise to a bounded operator $C_{\lambda}$ on $\ell^2(\mathbb{N})$ and $$\|C_{\lambda}\| \leq \|A\|^{\lambda}\cdot\|B\|^{1-\lambda},$$ where $\|\cdot\|$ denotes the operator norm.

Such an estimate seems to be closely related with interpolation of operators. However the usual interpolation theorems that I know are for ''the same'' operator acting on different spaces. Is there a version of interpolations that could directly prove the above statement (and perhaps in a more general setting)?

Another related question: which functions $\varphi(x,y)$ on two variables that give rise to the norm inequality $$\Big\|\Big(\varphi(a_{mn}, b_{mn})\Big)_{m,n}\Big\| \leq \varphi(\|A\|,\|B\|),$$ where $A=(a_{mn})$ and $B=(b_{mn})$ are two bounded operators on $\ell^2(\mathbb{N})$ whose entries are non-negative? We see above that $\varphi(x,y)=x^{\lambda}y^{1-\lambda}$ for $0\leq\lambda\leq 1$ works. In addition, $\varphi(x,y)=x^{s}y^{t}$ for non-negative integers $s,t$ also works and this follows from properties of Hadamard product.