# Hölder's inequality for matrices

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?

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Maybe a good rewording/related question is "Holder's inequality for function composition", i.e. take Holder's inequality and replace $fg$ with $f \circ g$. –  usul Feb 27 at 21:46
Thank you for the observation, @usul! Anyway, I don't know if an Hölder-like inequality for functions w.r.t. composition could imply something about my question in a straightforward way, because the $p$-norm of a function has a very different nature from the $p$-norm of a matrix... –  Paglia Feb 27 at 21:57
@usul I don't see why this would be a good idea. The question asks about an inequality involving the norm of a product of elements of an algebra (which is what H\"older's inequality also treats) and so composition of scalar-valued functions does not seem obviously relevant. –  Yemon Choi Feb 28 at 2:15
@Yemon,Paglia: agreed that it is not all that related, I just reacted without stopping to think about operator norms vs function norms and so on. –  usul Feb 28 at 2:36

## 1 Answer

The closest thing I know for induced norms is the Riesz–Thorin theorem. There are other Hölder-like inequalities for matrices, for example involving Schatten norms.

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