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The self-adjoint operator $A^*A$ (for a generic linear operator $A$) is used in functional analysis, linear algebra, statistics, physics, and probably many other fields. I am curious if there is a standard way to refer to this operator? There's the Gramian of course, but that seems specifically to refer to a collection of finite-dimensional vectors. "Gram operator" seems natural to me, but Googling suggests that it's not widely used.

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  • $\begingroup$ Gilbert Strang loves to emphasize the importance of $A^* A$ in linear algebra, and I don't recall him having a name for it. (He also emphasizes $A^* C A$, where $C$ is symmetric positive definite.) $\endgroup$ – littleO Jan 21 '17 at 21:19
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    $\begingroup$ I have seen the notation $A^G=A^*A$ to emphasize this "Gramian" operation ... as an operator. $\endgroup$ – Jean Marie Jan 21 '17 at 21:27
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In the context of Quantum Mechanics the operaror a*a with the same characreristis of your given operator is called the Number operator (in the framework of creation and anihilation operators). The number in this case accounts for the number of particles in a given quantum state.

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