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Pseudospectra and singular values of large convolution operators. (English) Zbl 0819.45002
Let \(\Lambda_ \varepsilon (T)\) be the pseudospectrum of linear bounded operator \(T\): \(\Lambda_ \varepsilon (T) = \{\lambda \in \mathbb{C}, T - \lambda I\) is not invertible or \(\| (T - \lambda I)^{-1} \| \geq 1/ \varepsilon\}\). The author investigates the connection between the pseudospectra of the Wiener-Hopf operator \(W = W_{0, \infty}\) of the truncated operator \[ (W_{0, \tau} \varphi) (x) = \int^ \tau_ 0 k(x - t) \varphi (t)dt, \quad 0 < x < \tau. \] For a large class of kernels \(k\) he shows that \(\lim_{\tau \to \infty} \Lambda_ \varepsilon (W_{0, \tau}) = \Lambda_ \varepsilon (W)\) for each \(\varepsilon > 0\).

MSC:
45C05 Eigenvalue problems for integral equations
45P05 Integral operators
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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