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Spectral resolutions for non-self-adjoint block convolution operators

Zalot Ewelina

Opuscula Mathematica

2022

Vol. 42, no. 3

459--487

The paper concerns the spectral theory for a class of non-self-adjoint block convolution operators. We mainly discuss the spectral representations of such operators. It is considered the general case of operators defined on Banach spaces. The main results are applied to periodic Jacobi matrices.

Comparison of direct and perturbation approaches to analysis of infinite-dimensional feedback control systems

Grabowski Piotr

International Journal of Applied Mathematics and Computer Science

2021

Vol. 31, no. 2

195--218

For general boundary control systems in factor form some necessary and sufficient conditions for generation of an analytic exponentially stable semigroup are proposed in both direct and perturbation forms for comparison. The direct approach is applicable to operators with the numerical range satisfying certain additional conditions. In particular, it applies to operators similar to convexoids and therefore it generalizes previous results known for hyponormal operators. The perturbation result (indirect approach) is derived and formulated as an exponential stability robustness result using the frequence-domain considerations. It is expressed in terms of some estimates of the resolvent growth over the open right complex half-plane and compared with some recent results. The analysis is illustrated in detail with examples of an unloaded and loaded electric RC-transmission line with proportional negative feedback.