I was wondering if the Hölder's inequality was true for matrix induced norms, i.e. if
$$\|AB\|\_1 \leq \|A\|\_p\|B\|\_q, \quad\forall p,q \in [1,\infty] \text{ s.t. } \tfrac{1}{p}+\tfrac{1}{q} = 1.$$
But it seems that this does not hold in general, in fact
$$A = \begin{bmatrix}1 & 2\\ 0 & 0 \end{bmatrix}, \; B = \begin{bmatrix}1 & 0\\ 2 & 0 \end{bmatrix}, \; p = 1, \; q = \infty $$
is a simple counterexample, and it is not hard to find similar ones for other choices of $p$ and $q$.


**Question** *Does a Hölder-like inequality hold for matrix induced norms?*



Hölder's inequality for matrices
There are (at least two) "generalizations" of Hölder inequality to the non-commutative case. One is the so called tracial matrix Hölder inequality:


$$
|\langle A, B \rangle\_{HS} |= |\mathrm{Tr} (A^\dagger B) | \le \| A\|\_p \,\, \| B\|\_q 
$$


where $\| A\|\_p$ is the Schatten $p$-norm and $1/p+1/q=1$. You can find a proof in [Bernhard Baumgartner, An Inequality for the trace of matrix products, using absolute values](http://arxiv.org/pdf/1106.6189v2.pdf).


Another generalization is very similar to what you wrote and reads


$$ \parallel|AB|\parallel \, \le\, \parallel |A|^p\parallel^{1/p} \,\, \parallel|B|^q \parallel^{1/q}
$$
where $|M|:=(M^\dagger M)^\frac12$
and it holds whenever $ \parallel \cdot \parallel$ is a unitarily invariant norm. You can find a proof in the book of Bhatia *Matrix Analysis*.


