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During my studying of many papers, some authors used what so-called $T T^*$ argument. I have no clue about this concept (or mathematical tool). Could you please enlighten me with some explanations or/and references! Thanks in advance.

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    $\begingroup$ CAn you give some references where you found it? $\endgroup$
    – András Bátkai
    Commented Apr 8, 2022 at 7:46
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    $\begingroup$ The best place to learn this is the paper by Ginibre, Velo "Generalized Strichartz Inequalities for the Wave Equation" Journal of Functional Analysis Volume 133, Issue 1, October 1995, Pages 50-68 $\endgroup$
    – Piero D'Ancona
    Commented Apr 8, 2022 at 8:31
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    $\begingroup$ To be honest, in that paper a rigorous statement is given (Lemma 2.2), which in my opinion is very clear, but no proof is given. However the proof is really simple, you could do it as an exercise after reading the statement $\endgroup$
    – Piero D'Ancona
    Commented Apr 8, 2022 at 8:33
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    $\begingroup$ The $TT^*$ idea goes back a long way! It is used, for example as the first step Tomas' proof of the restriction theorem in 1975 (Bull. AMS, vol 81, no. 2, p.477-478). It is important not so much in itself but as a problem solving strategy. Sometimes the operator $TT^*$ is just a bit easier to analyze than $T$ itself. $\endgroup$
    – Willie Wong
    Commented Apr 8, 2022 at 14:15
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    $\begingroup$ @PieroD'Ancona: I am also tempted to say that paper is almost the only place where the rigorous statement (including the density argument) is carefully formulated. (I spent some time looking for a precise statement when preparing my lecture notes last year.) $\endgroup$
    – Willie Wong
    Commented Apr 8, 2022 at 15:20

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