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Iterative solution of eigenvalue problems for normal operators. (English) Zbl 0708.65055
Let \(X\) be a complex Hilbert space, \(T\colon X\to X\) a linear bounded and normal operator. The following Kellog type iterations
\[ x^{(n+1)}=Tx^{(n)},\quad \mu_n=<x^{(n+1)}, y_n>/<x^{(n)}, z_ n>,\quad x_n=k_n^{-1}x^{(n)} \tag{1}\]
are studied, where the sequences \((y_n)\), \((z_n)\) of elements of \(X\) and the number sequence \((k_n)\) are chosen such that the denominators in (1) are not equal to zero and \(\lim_{n\to \infty}y_n=\lim_{n\to \infty}z_n=y\in X\). If \(x^{(0)}\) is a suitable starting approximation, then \(\lim_{n\to \infty}x_n=\bar x_0\), \(\lim_{n\to \infty}\mu_n=\bar \mu_0\), where \(\bar x_0\) is the eigenvector of \(T\) corresponding to the eigenvalue \(\bar \mu_0\).
Reviewer: J. Kolomý (Praha)

65J10 Numerical solutions to equations with linear operators
47A75 Eigenvalue problems for linear operators
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
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[1] N. Dunford J. T. Schwartz: Linear operators. I(II), Mir, Moskva 1962 (1966). · Zbl 0128.34803
[2] T. Kojecký: Some results about convergence of Kellogg’s iterations in eigenvalue problems. Czechoslovak Math. J.
[3] J. Kolomý: Approximate determination of eigenvalues and eigenvectors of self-adjoint operators. Ann. Math. Pol. 38 (1980), 153-158. · Zbl 0455.47015
[4] I. Marek: Iterations of linear bounded operators in non self-adjoint eigenvalue problems and Kellogg’s iteration process. Czechoslovak Math. J. 12 (1962), 536-554. · Zbl 0192.23701
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