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Extra resources for Analysis of Operators
Let us consider a domain Dom A∗ of the adjoint operator. Due to the closability of A∗ , its domain is a complete Hilbert space with respect to the graph-norm of A∗ . We denote this space by H+,∗ . The inner product in H+,∗ is given by (ϕ, ψ)+,∗ = (ϕ, ψ) + (A∗ ϕ, A∗ ψ), ϕ, ψ ∈ Dom A∗ . Clearly, H+,∗ is positive space with respect to H in the sense of , which means that · ≤ · +,∗ , H ❂ H+,∗ , where ❂ denotes the dense and continuous embedding of H+,∗ into H. It turns out that the domain Dom A of the operator A is a proper closed subspace in H+,∗ , though this set is dense in H.
The last means that there are no closed proper subspaces in H invariant under A. Without loss of generality, we can assume that A is a positive operator. Indeed, if the lower bound mA of the original operator is negative, then instead of A we take A = A + a1 (1 denotes the identity operator), where a is chosen, so that mA ≥ 0. Of course, it is necessary to make the inverse shift by −a1 after constructing and describing the self-adjoint extensions of the operator A . Moreover, it is convenient to assume that the lower bound mA = 1, namely A ≥ 1.
Conversely, a functional l deﬁned on G ⊂ L, is called the restriction of the functional ˆ l on L, if l(f ) = ˆl(f ) for all f ∈ G, and one notes ˆl G = l. Let us notice some important properties of linear functionals. 4. Let L be a linear normed space and G ⊂ L be a linear dense subset. Then for any continuous linear functional l deﬁned on G, there exists a unique continuous linear functional ˆl on L as an extension by continuity such that ˆl G = l. In addition, ˆl = l . 3. 5 (Hahn–Banach). Let L be a linear normed space and G ⊂ L be a linear subset of L.