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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 2nd 2017

    adjoint operator was in a very poor state, so I copied in at least the start of self-adjoint operator.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 26th 2023

    Before the line saying that adjoint operators need not exist in general, I added the sentence that on finite-dimensional Hilbert spaces they do.

    diff, v4, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeNov 16th 2023
    • (edited Nov 16th 2023)

    added (here) a quote by MacLane on the history of the notion:

    Two of von Neummann’s papers on this topic [Hilbert spaces] had been accepted in the Mathematische Annalen, a journal of Springer Verlag. Marshall Stone had seen the manuscripts, and urged von Neumann to observe that his treatment of linear operators TT on a Hilbert space could be much more effective if he were to use the notion of an adjoing T astT^ast to the linear transformation TT — one for which the now familiar equation

    Ta,b=a,T *b\;\;\;\;\; \langle T a, b \rangle \;=\; \langle a, T^\ast b \rangle

    would hold for all suitable aa and bb. Von Neumann saw the point immediately, as was his wont, and wishes to withdraw the papers before publication. They were already set up in type; Springer finally agreed to cancel them on the condition that von Neumann write for them a book on the subject — which he soon did [1932].

    This story (told to me by Marshall Stone) illustrates the important conceptual advance represented by the definition of adjoint operators. &lbrack…] I have written elsewhere [1970] that it is a step toward the subsequent description of a functor GG right adjoint to a functor FF, in terms of a natural isomorphism

    hom(Fa,b)hom(a,Gb)\;\;\;\;\; hom(F a, b) \;\simeq\; hom(a, G b)

    between hom-sets in suitable categories.

    diff, v5, current