# Spectrum of a special sum of matrices

Suppose I have a vector space with a tensor product structure $H = V^{\otimes m}$ and suppose I have some traceless Hermitian operator $A: V \otimes V \rightarrow V \otimes V$. Denote by $A_{i,j}$ the operator that acts as $A$ on the ordered $(i,j)$th tensor coordinates and as the identity everywhere else. I was wondering if there are any general tricks for either producing the spectrum or upper bounding the Schatten 1-norm of the operator

$$B = \frac{1}{m(m-1)}\sum\limits_{i \neq j} A_{i,j}.$$

I'm also particularly interested in the case in which the partial trace of A across either subsystem is the identity. Intuitively, I'd like to see how much the "frustration" of averaging misaligned operators reduces the spectrum of $B$. Here I suppose the parameters are $m, \dim(V)$, and $A$ itself. Of course if $A$ is the identity (and certainly not traceless), then there's no "frustration" and nothing happens, but what about more general $A$?

Thanks!