Essential Numerical Ranges of Linear Relations and Singular Discrete Linear Hamiltonian Systems

Abstract

In the paper, a concept of the essential numerical range W_e(T) of a linear relation T in a Hilbert space is given, other various essential numerical ranges W_ei(T), i = 1, 2, 3, 4, are introduced, and relationships among W_e(T) and W_ei(T) are established. These results generalize relevant results obtained by Bögli et al. in [Bögli, S., Marletta, M. and Tretter, C., The essential numerical range for unbounded linear operators, J. Funct. Anal., 279, 2020, 47–12]. Moreover, several fundamental properties of closed relations related to its operator parts are presented. In addition, singular discrete linear Hamiltonian systems including non-symmetric cases are considered, several properties for the associated minimal relations H₀ are derived, and the above results for abstract linear relations are applied to H₀.

Keywords

Linear relation / Numerical range / Essential numerical range / Essential spectrum / Singular discrete linear Hamiltonian system / 47A06 / 47A12 / 39A70

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