This paper deals with the infinite dimensional Hamiltonian operator with unbounded entries. Using the core of its entries, we obtain the conditions under which the numerical range of such an operator is symmetric with respect to the imaginary axis. Based on the symmetry above, a necessary and sufficient condition for generating $C_0$ semigroups is further given.
HUANG, JUNJIE; LIU, JIE; and CHEN, ALATANCANG
"Symmetry of numerical range and semigroup generation of infinite dimensional Hamiltonian operators,"
Turkish Journal of Mathematics: Vol. 42:
1, Article 6.
Available at: https://journals.tubitak.gov.tr/math/vol42/iss1/6