Fourth order eigenvalue problems with periodic and separated boundary conditions are considered. One of the separated boundary conditions depends linearly on the eigenvalue parameter λ. These problems can be represented by an operator polynomial L(λ) = λ 2 M -iαλK -A, where α > 0, M and K are self-adjoint operators.Necessary and sufficient conditions are given such that A is self-adjoint.MSC: Primary 34B07; secondary 34B08; 47E05
In the first part of the article perturbation of a closed form by a weakly continuous form is studied. This notion of weakly continuous perturbation is very handy and, as is shown in the article, leads to a new semigroup whose difference with the given semigroup consists of compact operators. We apply the results to elliptic operators on the Hardy space generalizing results from Semigroup Forum 95(2017), 281-292. A holomorphic semigroup operating on the Hardy space is obtained whose asymptotic behaviour is studied and which is compared with the semigroup generated by the elliptic operator with periodic boundary conditions on L2(0,2π).
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