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Let's consider Banach spaces $X,Y$ which are not Hilbert spaces and content (at least measurable) functions from $[0,1]$ to $\mathbb{R}.$

Is it possible to define Hilbert-Schmidt operators $\alpha\colon X \rightarrow Y$ or do $X$ and $Y$ have to be Hilbert spaces?

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    What you are looking for are nuclear operators of order $2$. See this page https://en.wikipedia.org/wiki/Nuclear_operator#On_Banach_spaces – 2017-02-07

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