Chapter 4 Rigged Hilbert Spaces A considerable part of functional analysis, including the theory of linear operators, particularly the spectral theory, cannot be presented successively without the no- tion of a rigged Hilbert space. The use of the method of rigged spaces allows one to beyond the setting with two starting objects, a single Hilbert space H and an operator A on H. To provide a complete picture of the spectral properties for A in the general case, it is necessary to equip the space H with an additional couple of Hilbert (or topological) spaces in a specific way. So, in a natural way there appear ∗ the triples of embedding spaces of a view H− ⊃H⊃H+, (or Φ ⊃H⊃Φ). In this book we will restrict our attention to the Hilbert equipping which form the so-called rigged Hilbert spaces. In the most advanced approach (see [42] and references therein) it is assumed that Dom A is a part of some positive Hilbert space H+, which is continuously embedded into H. Then the generalized eigen-functions of the operator A belong to the negative space H− which extends H. This approach proved its success in a wide class of various problems of mathematical physics, the theory of differential operators, and, especially, in the infinite-dimensional analysis (see, e.g., [45]). In this chapter we describe the standard construction of the rigged Hilbert space following to the detail presentation in Berezansky’s books [42, 44] (see also [48]). In addition, we will briefly analyze the properties of the A-scale of Hilbert spaces which will be used in that follows. 4.1 Construction of a rigged Hilbert space By definition (see [42, 44]), a triple of Hilbert spaces H− ⊃H0 ⊃H+ forms a rigged Hilbert space,ifH+ is a proper subset of H0, and, in turn, H0 is a proper subset of H−, and the following three conditions are fulfilled (these conditions are not independent): © Springer International Publishing Switzerland 2016 61 V. Koshmanenko, M. Dudkin, The Method of Rigged Spaces in Singular Perturbation Theory of Self-Adjoint Operators, Operator Theory: Advances and Applications 253, DOI 10.1007/978-3-319-29535-0_4 62 Chapter 4. Rigged Hilbert Spaces H ⊃H H ⊃H (a) both embedding 0 + and − 0 are continuous and dense, we de- note, H− = H0 = H+; (4.1) (b) the norms in H−, H0,andH+ satisfy the inequalities ·− ≤·0 ≤·+; (4.2) (c) the spaces H− and H+ are dual to one another with respect to H0. By conditions (a) and (b), the identity mappings H+ ϕ −→ ϕ ∈H0, H0 f −→ f ∈H− are continuous. Condition (c) means that for each vector ϕ ∈H+, the linear functional lϕ(f):=(f,ϕ)0,f∈H0 has an extension by continuity on the whole space H−. This extension defines the duality (inner) pairing between H− and H+: ω,ϕ −,+,ω∈H−,ϕ∈H+. It is clear that this duality pairing extends the usual inner product in H0, i.e., the following equality holds true: ω,ϕ −,+ =(f,ϕ)0, if ω = f ∈H0. In addition, the duality pairing between H+ and H− in the rigged Hilbert space (4.1), as an extension of the inner product (·, ·)0 in H0, satisfies the symmetry condition ∈H ∈H ω,ϕ −,+ = ϕ, ω +,−,ω −,ϕ +. Of course, the latter product also is continuous in ω ∈H− and ϕ ∈H+.Inpartic- ular, if we fix ω ∈H−,thenω,ϕ −,+ defines an conjugate-linear and continuous functional on H+. Therefore, the spaces H−, H+ are dual to one another with re- spect to H0. In accordance with [42], we name these spaces negative and positive, respectively. We note that the rigged space (4.1) can be constructed starting with a couple of embedded Hilbert spaces H and K if they constitute a pre-rigged space, i.e., if one of them is a dense subset of the other, for example H⊃K. In addition, the inequality ϕH ≤ϕK, ∀ϕ ∈K holds true. Hence, we can write H = K.WedenoteH = H0,andK = H+.Then ϕ0 ≤ϕ+, ∀ϕ ∈H+. 4.1. Construction of a rigged Hilbert space 63 Now we can introduce the negative norm f− := sup |(ϕ, f)0|,ϕ∈H+ ϕ +=1 for vectors f ∈H0. It is clear that this norm satisfies the inequality f− ≤f0, ∀f ∈H0. The completion of H0 with respect to the negative norm gives the space H−,which contains H0 and is dual to H+. Thus, beginning with a pre-rigging H0 = H+,we obtain a rigged Hilbert space of the form (4.1). It should be noted that starting with a pair H = K, we can put H = H−, K = H0, and construct a rigged space in a slightly different way. Specifically, the pre-rigging couple H− = H0 can be extended to the right side. Let us describe the corresponding procedure in more detail. We will construct the positive space H+ by means of the linear functionals lϕ(f):=(f,ϕ)0,f∈H0, defined for every fixed ϕ ∈H0. It is clear that lϕ(f) is continuous on H0.But, in general, it is not continuous on H−. We form the positive space H+ by taking only those ϕ ∈H0 for which the functional lϕ(f) has a continuous extension on the whole space H−. It is easy to show (see [42, 44]) that above ϕ’s form a linear, dense in H0 set. The positive norm ϕ+ is defined by the obvious formula ϕ+ =sup|(f,ϕ)0|,f∈H0. f −=1 The space H+ consisting of the vectors described above is complete with respect to this norm. The inner product in H+ is determined by the polarization identity: 1 (ϕ, ψ) = (ϕ + ψ2 −ϕ − ψ2 + iϕ + iψ2 − iϕ − iψ2 ). + 4 + + + + Clearly, ϕ0 ≤ϕ+. Thus, starting with H− = H0, we construct the rigged Hilbert space H− = H0 = H+. The important role in the theory of rigged spaces is played by the canonical identification operators, which we call the Berezansky canonical isomorphisms. They arise as follows. Let us consider for a fixed ϕ ∈H+ the functional lϕ(f)=(f,ϕ)0,f∈H0. It has a continuous extension to H− which can be written via the duality pairing: lϕ(ω)=ω,ϕ −,+. Now, according to the Riesz theorem, lϕ, as a functional on 64 Chapter 4. Rigged Hilbert Spaces ∗ ∗ H−, has the representation lϕ(ω)=ω,ϕ − with some ϕ ∈H−. It is understood ∗ that ϕ+ = ϕ −. Therefore, the mapping ∗ D−,+ : H+ ϕ −→ ϕ ∈H− (4.3) is isometric. The operators H −→ H −1 H −→ H D−,+ : + −,I+,− = D−,+ : − +, (4.4) are called the Berezansky canonical isomorphisms. It is a simple exercise to prove the validity of the following relations: ∗ ω,ϕ −,+ =(ω,ϕ )− =(ω,D−,+ϕ)− =(I+,−ω,ϕ)+, 2 ≥ ϕ, D−,+ϕ +,− = ϕ + 0, 2 I+,−ω,ω +,− = ω− ≥ 0, ∗ D−,+ϕ− = ϕ − = ϕ+,ω∈H−,ϕ∈H+. For further considerations it is important that each rigged space of the form (4.1) can be continued any number times in both directions, left and right. In particular, by the above-described procedure we can extend the rigged space (4.1) to a chain containing five spaces, H−− = H− = H0 = H+ = H++. (4.5) For example, to obtain H−−, we need to continue the pre-rigging couple H− = H0 to the rigged space H−− = H− = H0, where H−− is the dual to the space H0 space with respect to H−. In turn, the space H++ is dual to H0 with respect to H+. At the same time, H++ is dual to H−− with respect to H0. The indicated procedure can be iterated countably many times, yielding the discrete scale of Hilbert spaces ···= H−k = ···= H− = H0 = H+ = ···= Hk = ··· , (4.6) where k ∈ N, H+ := H1,andH++ := H2. Exercise 4.1.1. Show that a pair of Hilbert spaces H and K whose norms satisfy the inequality ϕH ≤ϕK, ∀ϕ ∈K, form a pre-rigging couple H = K only if the kernel of the identity mapping K ϕ → ϕ ∈His zero. Example 4.1.2. Let γ ≥ 0 be a positive, densely defined, and closed quadratic form on H. It is well known (see Chapter 2) that the domain Q(γ) of such a form is a complete Hilbert space with respect to the inner product (ϕ, ψ)+ =(ϕ, ψ)+γ(ϕ, ψ),ϕ,ψ∈ Q(γ). 4.2. Connections with self-adjoint operators 65 We denote this space by H+.Itisobviousthatϕ+ ≥ϕ for all ϕ ∈ Q(γ). It follows that H0 = H+,whereH0 ≡H. Thus, the pre-rigging couple H0 = H+ can be continued by means of the standard procedure described above to the rigged space H− = H0 = H+. Example 4.1.3. Let T be a closed linear operator on H with dense domain D(T ). Then H+ = D(T ) is a complete Hilbert space with respect to the inner product (ϕ, ψ)+ =(ϕ, ψ)+(Tϕ,Tψ),ϕ,ψ∈ D(T ). It is clear that H = H+ and these spaces form a pro-rigging couple which has an extension to the rigged space H− = H = H+. We can introduce the positive norm on D(T ) in a slightly different way. Namely, let us define an equivalent norm by ϕ+ = (1 + T )ϕ.ThenH− arises as the completion of H with respect to the −1 negative norm f− = (1 + T ) f, f ∈H. 4.2 Connections with self-adjoint operators There is a well-known connection between triplets of the kind (4.1) and positive self-adjoint operators in H0. The following theorem describes this connection in a general setting.