Let us regard the $n\times n$ matrices as operators on the $n$-dimensional $\ell\_p$ space; that is, we consider them as linear operators $\ell\_p^n\to \ell\_p^n$. When $p=2$, $M\_n$ is a C\*-algebra and we have 


$0 \leqslant A \leqslant B \implies \|A\|\leqslant \|B\|$.


Here $\|A\|$ denotes the operator norm of a map $A\colon \ell\_2^n\to \ell\_2^n$. What about other $p\in [1,\infty]$? 



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> Fix $p\in [1,\infty]$. Is it true that there exists $K>0$ such that for every $n$ and for all $A,B\colon \ell\_p^n\to \ell\_p^n$ with $0\leqslant A\leqslant B$ (meaning that $A$ and $B$ are self-adjoint and non-negative semi-definite) we have $$\|A\|\_{\ell\_p^n\to\ell\_p^n}\leqslant K\|B\|\_{\ell\_p^n\to\ell\_p^n}?$$
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My feeling is that it should be true for $p\in (1,\infty)$. For $p=1$ or $p=\infty$ there is an easy counter-example with $K=1$. Take 


$$A=\left[\begin{smallmatrix}2&1\\1& \tfrac{1}{2}\end{smallmatrix}\right],\;\;B = \left[\begin{smallmatrix}\tfrac{5}{2}&0\\0& \tfrac{5}{2}\end{smallmatrix}\right].
$$


Then $0\leqslant A\leqslant B$ yet for $p\in \{1,\infty\}$ we have $\|A\|\_{\ell\_p^2\to\ell\_p^2} = 3$ whereas $\|B\|\_{\ell\_p^2\to\ell\_p^2}=\tfrac{5}{2}$.


In the language of this thread: *[Monotone matrix norms](https://math.stackexchange.com/questions/374500/monotone-matrix-norms)*, I ask whether the operator $\ell\_p$-norms are monotone, that is, if we can take $K=1$. user147215 cleverly shows that this is not the case when $p\neq 2$.


**Possible approach**: It is not inconceivable that using some [Riesz–Thorin-type](http://en.wikipedia.org/wiki/Riesz%E2%80%93Thorin_theorem) argument we could show that the operator $\ell\_p$-norms are indeed monotone for $p$ is some neighbourhood of 2.



Order-preserving operator norms
The answer is no, Tomek. Use a Kashin decomposition of $L\_p^n$, $1\le p < 2$, to see that there are orthogonal projections whose norms as operators on $L\_p^n$ are of order $Cn^{|1/p-1/2|}$. (Kashin proved that for $1\le p < 2$ there is an orthogonal decomposition $A+B$ of $n$-space s.t. if $x \in A \cup B$, then $\|x\|\_p \le \|x\|\_2 \le C\|x\|\_p$, where I am using the uniform probability measure on $\{1,\dots,n\}$ rather than counting measure to define the norms.) 


The case $p>2$ follows by duality. 


B. S. Kashin, Sections of some
 finite-dimensional bodies and classes of smooth functions. 
 Izv. Acad. Nauk SSSR41 (1997), 334--351.


