# Integration on quasi-Banach spaces and Schatten ideals

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!

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 $$\ell^p$$.
• 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! – Curious Sep 30 at 16:39