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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)

##### MSC:
 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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##### References:
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