3
$\begingroup$

Let $[a,b]$ be an interval and $X$ a Banach space (for starters). We know that continuous functions $f:[a,b]\to X$ are Riemann integrable. Suppose now that $X$ is a quasi-Banach space, that is, its norm satisfies $\|x+y\|\leq K (\|x\|+\|y\|)$ for all $x,y\in X$ and some $K\geq 1.$

I found that, in general, quasi-Banach spaces (or $p$-Banach spaces) do not have this nice integrability property. Someone needs a notion of analyticity, see Albiac–Ansorena, 2013 (DOI link).

However, I am mainly interested in the case where $X$ is the $\mathcal{L}^p(H)$ Schatten *-ideal on a separable Hilbert space $H$, where $p\in (0,1)$. Note that for $p\geq 1$ it is a Banach space (Bnach $*$-ideal). Some facts about this quasi-Banach space (case $p\in (0,1)$).

  1. Every $T\in \mathcal{L}^p(H)$ is compact
  2. For $T\in \mathcal{L}^p(H)$ the quasi-norm is $\|T\|_p= \| s_n(T)\|_{\ell_p}$, where $(s_n(T))_{n\geq 0}$ is the sequence of singular values in decreasing order, counting multiplicity.
  3. For $A,B\in \mathcal{B}(H)$ and $T\in \mathcal{L}^p(H)$ it holds that $\|ATB\|_p\leq \|A\|_{\mathrm{op}}\|T\|_p\|B\|_{\mathrm{op}}$
  4. Clearly, for $T\in \mathcal{L}^p(H)$ we have $\|T\|_{\mathrm{op}}\leq \|T\|_p$ since $s_0(T)=\|T\|_{\mathrm{op}}$.

Question: Is every continuous function $f:[a,b]\to \mathcal{L}^p(H)$ Riemann integrable ?

Thank you!

$\endgroup$
4
$\begingroup$

No, there are such continuous functions, which are continuous with values in $\mathcal{L}^p(H)$ for any $p$ but such that $\int_a^b f$ (which is well defined in the Banach space $\mathcal{L}^1(H)$) does not belong to $\mathcal{L}^p(H)$ for any $p<1$.

An almost counter-example is given as follows on $H=\ell^2$. Take a countable partition $([a_n,a_n+1))_{n \geq 2}$ of $[a,b)$ where $a_n - a_{n+1} = \frac{1}{n (\log n)^2}$, and define $f=\frac{1}{\log n} e_{n,n}$ (the usual matrix units) on $[a_n,a_{n+1})$, and $f(b)=0$. Then $\int_a^b f = \sum_{n \geq 2} \frac{1}{n (\log n)^3}e_{n,n}$ does not belong to $\mathcal{L}^p(H)$ for any $p<1$.

Of course, the previous function is not continuous (the discontinuity points are $a_n$ for $n>2$, here the factor $1/\log(n)$ is important to ensure continuity at $b$), but adding a bit of room between the intervals, you can turn $f$ into a continuous function with the same properties.

Observe that the function takes values in the diagonal matrices, so it is a counterexample in $\ell^p$.

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ That's great! Thanks! Do you know if holomorphicity helps? I mean, if $f:U\to \mathcal{L}^p(H)$ is holomorphic (differentiable) for $p<1$, is it integrable along paths? From what I understand, if $f$ is analytic (locally a power series), then it will be integrable. But if it is just holomorphic I don't know. For Banach space valued functions, these two notions are the same. But one uses Hahn-Banach Theorem. Thanks! $\endgroup$ – Curious Sep 30 at 16:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.