We have recently proven in our functional analysis II lecture that the spectrum of an operator is contained in the closure of its numerical range. On the German wikipedia page for the numerical radius they say that Otto Toeplitz already investigated the numerical range in 1918. But who has proven this result (first)?


1 Answer 1


This is called the spectral inclusion theorem. Toeplitz's 1918 paper Das algebraische Analogon zu einem Satze von Fejér already has it for finite dimensional operators as Satz 4. This was before the language of operators and Hilbert spaces introduced by von Neumann in 1927-29, see Highlights in the History of Spectral Theory by Steen, so it is phrased in terms of eigenvalues and bilinear forms:

"Die Eigenwerte einer beliebigen Bilinearform C gehoren alle zu ihrem Wertvorrat". [The eigenvalues of any bilinear form belong to its value range.]

According to Gustafson and Rao's Numerical Range, p.8, for bounded operators in Hilbert space the theorem was first proved by Wintner in Zur Theorie der beschränkten Bilinearformen (1929), who has spectrum and Hilbert spaces, but does not use the term "operator" either. For unbounded operators, the spectral inclusion is false. Stone in Linear Transformations in Hilbert Space (1932) mentions that for normal operators the closure of the numerical range is the convex hull of the spectrum. A generalization to non-linear operators is due to Zarantonello (1967).

  • $\begingroup$ @ViktorGlombik Gustafson and Rao call it that in their monograph. If you google "numerical range spectral inclusion" top results are on point. $\endgroup$
    – Conifold
    Oct 14, 2019 at 7:47

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