Geodesic
A polynomial iteration for the spectral family of an operator
Glasgow mathematical journal, Tome 6 (1963) no. 2, pp. 65-69

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Let T be a bounded symmetric operator in a Hilbert space H. According to the spectral theorem, T can be expressed as an integral in terms of its spectral family {Eλ}, each Eλ being a certain projection which is known to be the strong limit of some sequence of polynomials in T. It is a natural question to ask for an explicit sequence of polynomials in T that converges strongly to Eλ. So far as the author knows, no complete solution of this problem has been given even when H has finite dimension, i.e. when T is a finite symmetric matrix. Since a knowledge of the spectral family {Eλ} of a finite symmetric matrix carries with it a knowledge of the eigenvalues and eigenvectors, a solution of the problem may have some practical value. 
Bonsall, F. F. A polynomial iteration for the spectral family of an operator. Glasgow mathematical journal, Tome 6 (1963) no. 2, pp. 65-69. doi: 10.1017/S2040618500034754
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[1] 1.Bonsall, F. F., A formula for the spectral family of an operator, J. London Math. Soc. 35 (1960), 321–333. Google Scholar | DOI

[2] 2.Riesz, F. and Sz.-Nagy, B., Leçons d'Analyse Fonctionelle (Budapest, 1952). Google Scholar

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