Theory and Application of a Class of Abstract Differential-Algebraic Equations

DAEs: Infinite-dimensional LTI Index-1 Case 3.1 Infinite-dimensional Linear Algebra We first collect some useful infinite-dimensional definitions and statements which will be used in this and the next section. Details and proofs of these statements can be found in standard functional analysis references such as [11], [26], [28] and [38]: Proposition 3.1. Let L(X,Y ) denote the space of bounded linear operators from the Hilbert space X into the Hilbert space Y . L(X) represents the space L(X,X). (i) The finite sum and composition of bounded linear operators is a bounded linear operator. Specifically, for T1, T2, . . . , Tn ∈ L(X,Y ), K = T1+T2+ · · ·+Tn =⇒ K ∈ L(X,Y ) and for Ti : Xi → Xi+1, K = TnTn−1 . . . T2T1 =⇒ K ∈ L(X1, Xn+1). (ii) Given a bounded linear operator T ∈ L(X,Y ), kerT is a closed linear subspace, i.e., kerT = kerT . (iii) An operator P is a projector if it is idempotent, i.e., P 2 = P . Furthermore, projectors are bounded i.e., P ∈ L(X), and both the kernel and image are closed linear subspaces. For the special case of an orthogonal projector, we have im P = ker(I − P ) and (I − P ) = (I − P ) ∈ L(X) with kerP ⊕ im P = X. (iv) A bounded linear operator T ∈ L(X,Y ) is one-to-one (i.e., injective) if and only if kerT = {0}. (v) A bounded linear operator T ∈ L(X,Y ) that is one-to-one and onto all of Y (i.e., bijective) has a bounded inverse T−1 : Y → X.