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

Abstract In the paper,a concept of the essential numerical range We（T） of a linear relation T in a Hilbert space is given,other various essential numerical ranges Wei（T）,i=1,2,3,4,are introduced,and relationships among We（T） and Wei（T） are established.These results generalize relevant results obtained by B?gli et al.in[B?gli,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 H0are derived,and the above results for abstract linear relations are applied to H0.