Perturbing the boundary conditions of the generator of a cosine family

Abstract

Let A be a densely defined, closed linear operator (which we shall call maximal operator) with domain D(A) on a Banach space X and consider closed linear operators L : D(A) → ∂X and : D(A) → ∂X (where ∂X is another Banach space called boundary space). Putting conditions on L and , we show that the second order abstract Cauchy problem for the operator A with A u = Au and domain D(A ) := {u ∈ D(A) : Lu = u} is well-posed and thus it generates a cosine operator function on the Banach space X.

Spectral theory for structured perturbations of linear operators

“…In this section we introduce a general setting which generalizes boundary perturbations of operators in the sense of Greiner, cf. [8], and then apply to it the theory developed in Section 2.…”

“…Diagram 2. We note that in [8] the operator Φ : X → ∂X has to be bounded and P = 0. Under the Assumptions 3.5 below, which cover unbounded Φ and P , the spectral properties of A Φ P can be studied using our results from Section 2.…”

“…We mention that the operator A Φ for bounded Φ ∈ L(X, ∂X) was already studied by Greiner in [8]. In case dim(∂X) < +∞, In Section 4 we consider a series of concrete applications, most of which t into the setting of the generic example above.…”

“…Our interest in operators A BC given by (1.1) is, among other, motivated by perturbations of the domain of operators in the spirit of Greiner, cf. [8]. For this reason, in Subsection 3 we rst apply our main abstract result Theorem 2.3 to this generic situation.…”