Birkhoff and Rota, in their book Ordinary Differential Equations (4e), claim on p.55 that:

The concept of the adjoint of a linear operator, which originated historically in the search for integrating factors, is of major importance because of the role which it plays in the theory of orthogonal and biorthogonal expansions.

I am interested in substantiations of the historical claim of Birkhoff and Rota; i.e. the claim that the roots of the notion of an adjoint is in the pursuit of finding integrating factors. It is admissible to interpret the word "adjoint" with arbitrarily specific scope. Different words signifying similar ideas are also admissible. (Consequently my question is not about terminology; compare What is the first usage of the term "Adjoint" and why was this word chosen?, The terminologies "Adjoint" and "Adjugate")

For context, this is how Birkhoff and Rota define the adjoint: for $L= p_2 D^2+p_1 D+p_0$ a linear differential operator,

  • $L$ is exact if for some $q_1,q_0$, $L = D(q_1 D + q_0)$,
  • Thus $L$ is exact iff $p_2''-p_1'+p_0=0$,
  • $v$ is an integrating factor for $L$ if $vL$ is exact,
  • Thus $v$ is an integrating factor iff $M[v]=(vp_2)''-(vp_1)'+(vp_0) = 0$,
  • $M = p_2D^2 + (2p_2'-p_1)D + (p_2''-p_1'+p_0)$ is the adjoint of $L$.

(Thus for some appropriately chosen functions, $\langle v,L[u]\rangle = \langle M[v], u\rangle$, the bilinear pairing being through an integral.)


1 Answer 1


Yes, Birkhoff and Rota make an unnecessarily cryptic allusion to Lagrange's memoir Solution de différents problèmes de calcul intégral (1766), where he multiplies a linear equation by a factor and then integrates by parts. This leads to what will later be called "adjoint equation" for the (integrating) factor, see Demidov, On the History of the Theory of Linear Differential Equations, 2.1.

The name "adjoint" was attached by Fuchs a century later in Zur Theorie der linearen Differentialgleichungen mit veränderlichen Coefficienten (1866). The transition from adjoint equation to adjoint operator/transformation (on function spaces with inner product given by an integral) was made by Riesz in Über lineare Funktionalgleichungen (1916). He used it to study operator inverses, see Bernkopf, The Development of Function Spaces, pp.58-62. Note that a different use of "adjoint" was around at the time as introduced by Bocher in 1907, for what is now called "adjugate".

  • $\begingroup$ Thanks! I will have a look at the papers you cited. $\endgroup$
    – Alp Uzman
    Dec 6, 2023 at 4:52

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