David E. Rowe’s fascinating paper [5] presents many facts and facets of the dramatic life and work of Fritz Noether. On the scientific side, Rowe’s paper is focused on research in physics, especially in hydrodynamics. Remembering Fritz Noether would be incomplete, however, without mentioning his fundamental contribution to operator theory. In his 1920 paper [4] on singular integral equations, Noether was the first to describe a relation between the functional-analytic “index” of an operator and the topological index of its “symbol” (these terms will be defined below). This pioneering work marks the beginning of index theory, which culminated in the famous Atiyah–Singer index theorem.

Noether’s paper was motivated by a problem in the theory of tides, and in it he announced applications to viscous fluid flows to appear in a future paper.Footnote 1 The physical problem was modeled by a singular integral equation involving a kernel that has a pole of first order on the contour of integration, so that the integral is to be understood as a Cauchy principal value. This integral equation is related to a boundary value problem for analytic functions on the complex unit disk, belonging to a large class nowadays referred to as Riemann–Hilbert problems. Problems of this type were studied before Noether by several authors, including David Hilbert, but all of them missed a crucial fact.

Recall that the functional-analytic index of a linear operator A is defined as the difference of the numbers of linearly independent solutions (assumed to be finite) to the homogeneous equation \(Ax=0\) and its adjoint \(A^*x=0\),

$$\begin{aligned} {\text {ind}}A := {\text {dim}}\,{\text {ker}}A - {\text {dim}}\,{\text {ker}}A^*. \end{aligned}$$
Figure 1:
figure 1

Reproduction of Noether’s main result from his paper [4].

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In his paper [4] in the Mathematische Annalen, Noether introduced this quantity and discovered the fundamental result shown in Figure 1. Note that the number n has a geometric interpretation: the function \(a-{i}b\) (later called the symbol of the equation or the operator) maps the unit circle into the complex plane, and n is its winding number, i.e., the number of oriented turns of the image curve around the origin.

Noether pointed out in two footnotes of [4] that David Hilbert had made a mistake in his talk [1] at the third International Congress of Mathematicians and the subsequent paper [2]. These footnotes are reproduced in Figure 2 and translated here:

6) In this respect, the Heidelberg talk by D. Hilbert quoted above contains an error; cf. below in §1, Rem. \(^{10})\).

10) This second step of the proof is missing in Poincaré (loc. cit.) and in Kellog (loc. cit.) (except in a special example ...). In contrast, it is begun by Hilbert (...); however, the implementation is not correct, because it contains in essence the incorrect assumption that the homogeneous equations \(K(\omega ) = 0\) and \(\overline{K}(\omega ) = 0\) would have the same number of solutions ....

Figure 2:
figure 2

Footnotes 6 and 10 from Noether’s paper [4] with the criticism of Hilbert.

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Since Hilbert (together with Felix Klein, Albert Einstein, and Otto Blumenthal) was an editor of the Mathematische Annalen, he was very likely aware of Noether’s criticism, but his immediate reaction seems not to be known. In his book [3], he circumvented the critical part of his “proof” and rephrased the result in a formally correct (but vague) form, still avoiding the index. And he did not mention Noether’s paper.

In hindsight of its importance, the moderate number (currently less than 50) of citations of [4] is somewhat surprising. This may be due to the fact that in his time, Fritz Noether was known as an expert in hydrodynamics rather than in operator equations. Another reason is certainly his difficult life and tragic death, as impressively documented by David Rowe [5]. Last, but not least, that Hilbert chose to pretend ignorance of Noether’s observation certainly had a negative effect on the recognition of his results.

In contemporary terminology, operators A having closed range and finite-dimensional kernels of A and \(A^*\) are often referred to as “Fredholm operators.” Since Fredholm’s work (on integral equations) covers only operators with index zero, it would be a late, but historically accurate, compensation to speak of “Noether operators,” at least if their indices are different from zero.