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(This answer expands the ideas in the comments by Mikael de la Salle and myself). There is a family of matrix norms that can be defined as follows: given $p \geq 1,k \leq n$, $$\|A\|_{p,k} = \left(\sum_{i=1}^k \sigma_i(A)^p\right)^{1/p},$$ where we denote by $\sigma_i(A)$ the $i$th singular value of $A$ (taken in nonincreasing order, $\sigma_1(A)\geq ... 2 Let$A$be a square matrix of dimension$n\times n$and consider the following norm for$1< p<\infty$: $$\|A\|_{p} = \max_{x \neq 0} \frac{\|Ax\|_p}{\|x\|_p}.$$ Let$\psi_p(x):=\big(|x_1|^{p-1}\operatorname{sign}(x_1),\ldots,|x_n|^{p-1}\operatorname{sign}(x_n)\big)$and write$p'$the Hölder conjugate of$p$. Then for every$x\$ we have $$\max_{x\neq ... 4 The following argument seems easier, but there might be a still more fundamental one. Notice that  \phi: A^{\sim} \to \mathbb{C}  above is also a  C^{*} -algebraic homomorphism. As  C^{*} -algebraic homomorphisms are automatically contractive (which is a consequence of a not-too-difficult spectrum argument), we have$$ \forall (a,\lambda) \in A \times ...