Integration on quasi-Banach spaces and Schatten ideals

Question

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!

Answer 1

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 $\ell^p$.