Approximation in eigenvalue problems for holomorphic Fredholm operator functions. II: Convergence rate. (English) Zbl 0880.47010
Summary: We prove some asymptotic error estimates for the difference of eigenvalues of a holomorphic operator function and its approximations. We assume the discrete approximation scheme of F. Stummel [Math. Ann. 190, 45-92 (1970; Zbl 0203.45301)] for spaces and the regular approximation scheme for operator functions.
To get error estimations two approaches are used in this paper. In Sections 2.1 and 2.2 the problem is reduced to the case of matrix functions by the construction, presented in part I [cf. review above]. Here we follow the approach of earlier papers by the author. Some notations and results of part I are briefly recalled in Paragraph 1.
In Section 2.3, the estimation is derived by transforming of an appropriately chosen identity. Here we follow papers by G. Vainikko and the author.
To get error estimations two approaches are used in this paper. In Sections 2.1 and 2.2 the problem is reduced to the case of matrix functions by the construction, presented in part I [cf. review above]. Here we follow the approach of earlier papers by the author. Some notations and results of part I are briefly recalled in Paragraph 1.
In Section 2.3, the estimation is derived by transforming of an appropriately chosen identity. Here we follow papers by G. Vainikko and the author.
MSC:
| 47A75 | Eigenvalue problems for linear operators |
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 47A53 | (Semi-) Fredholm operators; index theories |
| 65J15 | Numerical solutions to equations with nonlinear operators |
Keywords:
asymptotic error estimates; difference of eigenvalues of a holomorphic operator function and its approximations; discrete approximation schemeReferences:
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