Rigged Hilbert Spaces

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 notion 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 denote, H− = H0 = H+; (4.1)

(b) the norms in H−, H0,andH+ satisfy the inequalities

·− ≤·0 ≤·+; (4.2)

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 respect 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. ∗ Theorem 4.2.1. Each unbounded self-adjoint operator A = A ≥ 1 on H0 is uniquely associated with the rigged Hilbert space

H−(A) = H0 = H+(A) (4.7) where the space H+ = H+(A) coincides with the domain D(A), equipped with the norm ϕ+ = Aϕ0, ϕ ∈ D(A). Conversely, for each rigged Hilbert space of the form (4.1), there exists the ∗ self-adjoint operator A = A ≥ 1 on H0 such that the triple (4.7) coincides with (4.1).

Proof. Let us start with an abstract rigged Hilbert space

H− = H0 = H+.

Consider the Berezansky canonical isomorphism

∗ D−,+ : H+ ϕ −→ ϕ ∈H−.

Its restriction to H0 we denote by LA, i.e.,

LA := D−,+ D(LA)={ϕ ∈H+ | D−,+ϕ ∈H0}. 66 Chapter 4. Rigged Hilbert Spaces

We claim that LA is a symmetric operator on H0. Indeed, for all vectors ϕ, ψ ∈ D(LA) ⊂H+ we have

∗ (LAϕ, ψ)0 =(D−,+ϕ, ψ)0 = ϕ ,ψ −,+ =(ϕ, ψ)+ ∗ (4.8) = ϕ, ψ +,− =(ϕ, D−,+ψ)0 =(ϕ, LAψ)0,

∗ ∗ where ϕ = D−,+ϕ and ψ = D−,+ψ.Infact,LA is self-adjoint on H0 since by construction, its range coincides with the whole space H0, i.e., RanA = H0.Itis 1/2 obvious, LA is unbounded and positive. So, we can deﬁne the operator A := LA . From (4.8) it follows that

1/2 1/2 (LA ϕ, LA ψ)0 =(Aϕ, Aψ)0 =(ϕ, ψ)+.

Therefore, D(A)=H+ and A ≥ 1since·+ ≥·0. Conversely, let A = A∗ ≥ 1 be an unbounded self-adjoint operator with the domain D(A)onH0. Starting with A we can to construct the rigged Hilbert space (4.7). To this end we notice that the domain D(A) forms a complete Hilbert space with respect to the inner product

(ϕ, ψ)+ := (Aϕ, Aψ)0,ϕ,ψ∈ D(A).

We denote this space by H+(A). Further, since H0 = H+(A), we can continue this chain to the rigged space (4.7). The corresponding procedure was described above. Now we can consider the Berezansky canonical isomorphism D−,+(A)in (4.7). It is clear that the operator LA related to D−,+(A)coincideswithA.Thus, we proved that there exists a bijective connection between rigged Hilbert spaces (4.7) and self-adjoint unbounded operators A ≥ 1onH0.

The next two propositions describe the above connection more precisely. Proposition 4.2.2. Given A as in Theorem 4.2.1, we consider the pre-rigging couple H− = H0,whereH− is the completion of H0 with respect to the norm

−1 h− = A h0,h∈H0. (4.9)

Then the positive space H+ corresponding to H− = H0 coincides with D(A) in the norm f+ := Af, i.e., H+ = H+(A).

Proof. According to the deﬁnition of the positive space, H+ consists of those f ∈H for which the linear continuous functional lf (h)=(h, f)0, h ∈H0 has an extension by continuity on the whole space H−. The latter requirement is equivalent to the property

f+ =sup|lf (h)| < ∞. h −=1 4.2. Connections with self-adjoint operators 67

In other words, it means that f belongs to D(A). Indeed, by the construction of the rigged Hilbert space, we have

lf (h)=(h, f)0 = h, f −,+ = D−,0ϕ, f −,+ =(Aϕ, f)0,ϕ∈ D(A).

Since D−,0 is isometric, the continuity of the functional lf (h)inh ∈H− is −1 equivalent to the continuity of this functional in ϕ = A h ∈H0. Therefore, ∗ ∗ ∗ ∗ ∗ lf (h)=(ϕ, f )0. This means that f ∈ D(A )andf = A f.SinceA = A ,we get f ∈ D(A). Thus, H+ = H+(A). ∗ Proposition 4.2.3. Given A = A ≥ 1 on H,letH0 = H+ be the pre-rigging couple, where H+ = D(A) with respect to the norm f+ := Af0. Then the negative space H− corresponding to H0 = H+ coincides with the completion of H0 with respect to the norm h−, i.e.,

H− = H−(A).

Proof. According to the construction of the rigged space, the negative space H− is deﬁned as the completion of H0 with respect to the norm

h− := sup |(f,h)0|,h∈H0. f +=1 We have to verify that this norm is equivalent to (4.9). Really, since A is self-adjoint and Ran A = H0,wecanwriteforeachh ∈H0:

(f,h)0 =(f,Aϕ)0 =(Af, ϕ)0 =(g,ϕ)0 with some ϕ ∈H+. Hence, −1 h− =sup|(f,h)0| =sup|(g,ϕ)0| = ϕ0 = A h0, f +=1 g 0=1 whereweusedtheequalityg0 = Af0 = f+ with g = Af.

Exercise 4.2.4. Construct H0 starting with a pre-rigging couple H− = H+. n Example 4.2.5. Let H = L2(R ,dx)andA be the multiplication operator deﬁned by a real-valued function. For example,

(Af)(x)= ρ(x)f(x),ρ(x)=1+|x|2, n 2 2 n where |x| := |xi| , x =(x1,...,xn) ∈ R . Then the triplet i=1 n −1 n n L2(R ,ρ dx) = L2(R ,dx) = L2(R ,ρdx) forms the rigged Hilbert space associated with the operator A. 3 Example 4.2.6. Let A =1− ΔonH = L2(R ,dx), where − Δ is the Laplace H H 2 2 R3 operator. Then + = +(A) coincides with the Sobolev space W2 = W2 ( ) H H −2 R3 (see Subsection 1.2.7) and − = −(A)=W2 ( ). Thus, one arrives at the well-known rigged space −2 = = 2 W2 L2 W2 . 68 Chapter 4. Rigged Hilbert Spaces

4.3 A-scales of Hilbert spaces

In what follows we will use the doubly-inﬁnite chain of Hilbert spaces associated with an operator A. Such a chain generalizes the rigging (4.6) and is called the A-scale. Below we describe its construction. Let A = A∗ ≥ 1 be a self-adjoint unbounded operator with the domain D(A) in the complex separable Hilbert space H with the inner product (·, ·). For each k ≥ 0 we deﬁne the Hilbert space Hk ≡Hk(A) which coincides as a k/2 set with the domain D(A ) equipped with the norm ·k corresponding to the inner product

k/2 k/2 k/2 (ϕ, ψ)k := (A ϕ, A ψ),ϕ,ψ∈ D(A ). (4.10)

We put H0 ≡Hand deﬁne H−k as the completion of H0 with respect to the negative norm ·−k generated by the inner product

−k/2 −k/2 (f,g)−k := (A f,A g),f,g∈H. (4.11) It is easy to see that for each ﬁxed k>0 the triplet

H−k = H0 = Hk (4.12) is a rigged Hilbert space in the sense of Berezansky’s deﬁnition [42, 44]. The chain of Hilbert spaces inﬁnite in both sides

···= H−k = ···= H0 = ···= Hk ··· (4.13) is called the A-scale. We will denote it by {Hk}k∈R. Example 4.3.1. A typical example of a scale of Hilbert spaces is provided by the Sobolev scale −k = = k W2 L2 W2 ,k>0, − ±k which can be considered as the A-scale for the operator A =1 Δ. The spaces W2 are deﬁned by the powers of the operator A =1−Δ or by their Fourier transforms, which are the operators of multiplication by (1 + |x|2)±k (see Subsection 1.2.7).

4.3.1 Properties of the A-scale The infinite chain of spaces (4.13) has a series of interesting properties. We will briefly discuss some of them, for more details, see [2, 51, 58, 143]. An important property is the invariance of the structure of rigged triplet (4.12) under shifts along the A-scale, i.e., shifts of the index k. We will call this property the first invariance principle of the A-scale. Its essence is that for any fixed k>0 and an arbitrary q, the triple of spaces, H−k+q , Hq, Hk+q form a new rigged Hilbert space, i.e., we can write

1 H−k+q = Hq = Hk+q,q∈ R . (4.14) 4.3. A-scales of Hilbert spaces 69

This property follows from the general procedure of construction for the rigged Hilbert space applied to one of the pre-rigging couples, Hq = Hk+q or H−k+q = Hq, taken from the A-scale. It should be remarked that by using the first invariance principle of the A- scale, we can extend a notion of the Berezansky canonical isomorphism for each ordered pair of spaces Hk and Hl, k>land define the operator Dl,k : Hk →Hl. −1 We denote Ik,l := Dl,k . It is clear that these operators are unitary mappings from Hk to Hl and from Hl to Hk, respectively. The next important property we call the second invariance principle of the A-scale. It essentially asserts that A is unitarily equivalent to its image under any shift produced by the operator Dl,0 or I0,l,l>2 along the A-scale, i.e., under the transition to any space in the scale. Let us explain this principle in more detail. To this end, we fix k ≥ 0and define the operator Ak := Dk,k+2. It is easy to check that Ak coincides with the restriction of A to Hk.So,Ak is self-adjoint in Hk+2. In accordance with this definition,

A ≡ Ak=0 = D0,2. In a similar way we deﬁne the operator

A−k := D−k,2−k,k>0.

It is self-adjoint on H2−k and coincides with the closure of A in H−k.Itisimportant ∈ R1 that all operators A±k, k +, are unitary images of the original operator A on H0.Inparticular,

Ak = Ik−2,0AD2,k,A−k = D−(k+2),0AI2,−k.

At the same time, the operator A is essentially self-adjoint in each space H−k, k>0. Thus, the second invariance principle of the A-scale can be formulated as follows. For any α ∈ R the operator Aα := Dα,α+2 is self-adjoint in Hα and is unitarily equivalent to the original operator A. It is obvious that if α>0, then Aα coincides with the restriction of A to Hα+2,andifα<0, then Aα is the closure of A in Hα+2. Moreover, it is easy to see that Aα is essentially self-adjoint in each space Hβ, with β<α. It is convenient for the sequel to recall once more the construction the A-scale and rewrite its properties in slightly diﬀerent notations. Starting with a self-adjoint operator A = A∗ ≥ 1, we deﬁne the Hilbert space Hα ≡Hα(A)foreachα>0. This space coincides, as a set, with the domain α/2 D(A ), and is equipped with the norm ·α corresponding to the inner product

α/2 α/2 α/2 (ϕ, ψ)α := (A ϕ, A ψ)0,ϕ,ψ∈ D(A ). (4.15) 70 Chapter 4. Rigged Hilbert Spaces

The space H−α is obtained as the completion of H0 with respect to the negative norm ·−α generated by the inner product

−α/2 −α/2 (f,g)−α := (A f,A g)0,f,g∈H0. (4.16)

The chain of spaces

···= H−α = ···= H0 = ···= Hα = ··· (4.17) with dense embeddings forms the A-scale of Hilbert spaces {Hα}α∈R.Itiseasyto see that for each ﬁxed α>0 the triplet

H−α = H0 = Hα (4.18) forms a rigged Hilbert space. Let H −→ H −1 H −→ H D−α,α : α −α,Iα,−α := D−α,α : −α α denote the Berezansky canonical isomorphisms. In accordance with the previous considerations, α/2 cl α/2 D−α,α =(A ) (A ), α/2 where cl denotes a closure of the mapping A : H0 →H−α/2. And what is more, we can introduce the Berezansky canonical isomorphisms between arbitrary pair of spaces Hα, Hβ from the A-scale: H −→ H −1 H −→ H Dα,β : β α,Iβ,α = Dα,β : α β,α<β. (4.19) For the further applications we list the following equalities which hold in the A- scale: D0,2 = A : H2 −→ H 0, −1 I2,0 = A : H0 −→ H 2; α/2 D0,α = A : Hα −→ H 0, α/2 cl D−α,0 =(A ) : H0 −→ H −α,α>0; (4.20)

D−α,β = D−α,0D0,β : Hβ −→ H −α,α,β>0;

(f,ϕ)0 = f,ϕ −α,α,f∈H0,ϕ∈Hα,α>0;

D00 = I00 ≡ 1.

In particular, putting H−α = H− and Hα = H+, α>0, we have

ω,ϕ −,+ =(ω,D−,+ϕ)− =(I+,−ω,ϕ)+,ω∈H−,ϕ∈H+, (4.21) where D−,+ ≡ D−α,α I+,− ≡ Iα,−α, and ·, · −,+ ≡·, · −α,α 4.3. A-scales of Hilbert spaces 71 denotes the duality pairing between positive and negative spaces. It should also be noted that

ω,ϕ −α,α = ω,ϕ −β,β,ω∈H−α,ϕ∈Hβ, 0 ≤ α<β, (4.22) although in general ω,ϕ −α,β = ω,ϕ −β,α. Of course, when the value of index α is not important, we denote the triplet (4.18) as (4.1). The next theorem follows from the previous considerations. Theorem 4.3.2. All four mappings listed below coincide with the self-adjoint oper- k/2 ators A , k>0 on H0 (or with their inverse):

(a) D0,k : Hk −→ H 0, (b) D−k/2,k/2 {f ∈Hk/2 | D−k/2,k/2f ∈H0}, (c) D−k,0 {f ∈H0 | D−k,0f ∈H0}, −k/2 (d) A = I0,−k {ω ∈H−k | I0,−kω ∈Hk}.

Proof. (a) is true according to the construction of the A-scale. Indeed, D0,k and k/2 A obviously coincide as operators acting from Hk into H0.Toprove(b)it is suﬃcient to remark that the operator D0,k can be deﬁned as a restriction of D−k/2,k/2.Thatis,

D0,k = D−k/2,k/2 {f ∈Hk/2 | D−k/2,k/2f ∈H0}.

In particular, D0,2 = A = D−1,1 {f ∈H1 | D−1,1f ∈H0}

= D−2,0 {f ∈H0 | D−2,0f ∈H0}.

The validity of (c) follows from the second invariance principle. Therefore, D0,k coincides with the restriction D−k,0 to Hk:

D−k,0 Hk = D0,k.

Hence, it holds that k/2 A = D−k,0 Hk. −1 Finally, (d) holds true due to the deﬁnition of the mapping I0,−k = D−k,0.Bythe second invariance principle, I0,−k H0 = Ik,0. Now it is obvious that

−1 −k/2 −1 Ik,0 = Dk,0,A = D0,k.

−k/2 Hence, A = I0,−k H0.Moreover,

−1 A = I0,−2 {ω ∈H−2 | I0,−2ω ∈H0}.