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The physicist Yoichiro Nambu introduced in a 1950 paper A Note on the Eigenvalue Problem in Crystal Statistics the notion of an "eigenoperator" (page 12, see Nambu and the Ising model for a recent discussion of this early work by the 2008 Nobel laureate).

Given a self-adjoint operator $H$, the eigenoperator $X$ satisfies $$HX-XH=\lambda X,\;\;\lambda\in\mathbb{R}.$$

Q: A Google search for "eigenoperator" does not return much, has this notion found its place in the mathematical literature, perhaps under a different name?

I append a screenshot of the relevant paragraph from Nambu's paper:

Footnote 6) refers to Nambu's paper On the Method of the Third Quantization, where $X$ is called an "eigenmatrix".
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Such an $X$ is an eigenvector of $\,\operatorname{ad}(H)$. Joint eigenspace decompositions of several $\operatorname{ad}(H_i)$ are commonplace in math since the work of Lie, Killing, Cartan, with the joint eigenvectors called root vectors. So I would say that the notion “had a place” already before Nambu.

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    $\begingroup$ Indeed, although probably not as easily describable before the gradual acceptance of a more abstract style post-Bourbaki...? .... including various levels of "(mathematical) reification" (=making an operation on things a thing in itself, that perhaps could be operated upon...) $\endgroup$
    – paul garrett
    Commented Sep 16, 2022 at 22:21
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    $\begingroup$ @paulgarrett You are right that one may perhaps not find $\text{ad}(\cdot)$ before Chevalley and Eilenberg [1941, 1948], nor root vector before Wang and Dynkin [1949, 1952a, 1952b]. $\endgroup$
    – Francois Ziegler
    Commented Sep 17, 2022 at 6:19

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