Given a sequence of square matrices $\lbrace A_n \rbrace$ over $\mathbb{C}$ of increasing dimension $d_n$, we say that the sequence of matrix sequences $\lbrace \lbrace B_{n,m} \rbrace_n \rbrace_m$ is an approximating class of sequences (a.c.s.) for $\lbrace A_n \rbrace$ provided that $$ A_n = B_{n,m} + R_{n,m} + N_{n,m} $$ with $R_{n,m}$, $N_{n,m}$ sequences of matrix of appropriate dimension such that $$ rk(R_{n,m}) \leq c(m) d_n, \quad \vert\vert{N_{n,m}}\vert\vert \leq \omega(m) $$ ($\vert\vert \cdot \vert\vert$ indicates the spectral norm) with $$ c(m) + \omega(m) \underset{m \to \infty}{\rightarrow} 0$$ (see for instance https://www.sciencedirect.com/science/article/pii/S0024379510005501).
My question is, if I change the norm through which I evaluate how $R_{n,m}$ is "small", will $B_{n,m}$ remain an a.c.s. for $\lbrace A_n\rbrace$?
Due to the fact that induced norm are equivalent (and the spectral norm is indeed induced), I suppose that the answer is affirmative for such norms.
However, what about Shatten-p norms or in general non-induced norms? Is the a.c.s. notion norm-agnostic?
Thank for any help