Meaning of this differentiation operators

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I have been just reading this paper here: paper and was wondering how they carry out the differentiation in (4.9). In principle, this should be just the differentiation of 4.8 with the help of 4.7a. But I don't get what this actually means and how this operator $T$ acts on 4.8? Is here anybody who can provide me with some details?

If anything is unclear, please let me know.