I read a statement in a paper and I cannot understand why it is true. Let $A,B$ be symmetric real matrices of possibly different sizes, with eigenvalues $(\lambda_k)_k$ and $(\gamma_j)_j$. Then it is known that the eigenvalues of $A \otimes B$ are $(\lambda_k \gamma_j)_{k,j}$.
Now, the paper says that as a consequence $$ ||A \otimes B||_{op} \leq ||A||_{op} \cdot ||B||_{op} $$
In my opinion, this should be an equality, since
if $A,B$ are symmetric then so is $A \otimes B$;
for a symmetric matrix $A$ the operator norm (with respect to $l_2$) is the spectral radius.
Where am I wrong? (Of course I know that if $=$ holds so does $\leq$, but then the authors would have written $=$ in the statement).