I am wondering whether the following sentence is punctuated properly:
From the previous section, the operator A is skew-adjoint; has compact resolvents; and thus has the required conditions.
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No. All the semicolons should be commas. See this wonderfully handy resource, The Punctuation Guide.
This sentence doesn't contain any of these constructions.
It would still be an awkward sentence, though. It might be clearer like this:
"From the previous section, the operator A has the required conditions because it is skew-adjoint and has compact resolvents."