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ME(EE) 550 Foundations of Engineering Systems Analysis
Chapter 06: Bounded Linear Operators on Hilbert Spaces
While some authors have used the terms ”Operator” and ”Transformation” synonymously, others specifically restrict operators as those transformations that map a vector space into itself. Linear transformations, that map a vector space V (over a field F) into another vector space W (over the same field F) will be denoted as L(V, W ) and bounded linear transformations as BL(V, W ). This chapter focuses on bounded linear operators on Hilbert spaces that will be denoted as BL(H, H), where H is a Hilbert space. In Chapter 05, we have shown that the dual space ℓ⋆ 2 is isometrically isomorphic to the ℓ 2 space itself. By Reisz-Frech´et Theorem, introduced and proved in Chapter 05, it follows that any Hilbert space is isometrically isomorphic to its dual space. In this chapter, we will show that there exists an isometric isomorphism between any two Hilbert spaces, defined over the same field, provided that they have the same Hilbert dimension. This chapter should be read along with Chapter 4 and Chapter 5 of Naylor & Sell. Specifically, some of the solved examples and exercises in Naylor & Sell would be very useful.
1 Basic concepts
It follows from Reisz-Frech´et Theorem that ∀y ∈ H there exists a unique functional y⋆^ ∈ H⋆^ such that y⋆(x) = 〈x, y〉 ∀x ∈ H. Now, let T ∈ BL(H, H) be a bounded linear operator mapping a Hilbert space H into itself such that 〈T x, y〉 = 〈x, y⋆〉. We introduce the adjoint operator T ⋆^ : H → H defined by 〈T x, y〉 = 〈x, y⋆〉.
Definition 1.1. (Adjoint operator) Let H be a Hilbert space and let T ∈ BL(H, H). Then, T ⋆^ : H → H, called the (Hilbert-) adjoint operator of T , is defined as
〈T x, y〉 = 〈x, T ⋆y〉 ∀x, y ∈ H
We also denote y⋆^ = T ⋆y ∀y ∈ H, where y⋆^ is a bounded linear functional on H, i.e., y⋆^ ∈ H⋆.
Proposition 1.1. (Uniqueness of the adjoint operator) Let H be a Hilbert space. Then, the adjoint operator T ⋆^ is unique for every operator T ∈ BL(H, H),
Proof. The proof follows from Riesz-Frech`et Theorem (see Chapter 05).
Remark 1.1. The following identities are derived directly from Proposition 1.1.
(i) I⋆^ = I, where I is the identity operator, i.e., Ix = x ∀x ∈ H.
(ii) 0⋆H→H = (^0) H→H , where (^0) H→H is the zero operator on H, i.e., (^0) H→H x = (^0) H ∀x ∈ H.
(iii) (αT )⋆^ = α T ⋆^ ∀T ∈ BL(H, H) and every scalar α.
(iv) (S + T )⋆^ = S⋆^ + T ⋆^ and (ST )⋆^ = T ⋆S⋆^ ∀S, T ∈ BL(H, H).
Theorem 1.1. Let H be a Hilbert space and let K ∈ BL(H, H). Then,
(i) K⋆^ ∈ BL(H, H).
(ii) K⋆⋆^ = K.
(iii) ‖K⋆‖ = ‖K‖.
Proof. (i) Linearity: Let H be a Hilbert space over the field F. Let x, y, z ∈ H and α, β ∈ F. Then, 〈x, K⋆(αy+βz)〉 = α〈x, K⋆y〉+β〈x, K⋆z〉 = 〈x, K⋆αy〉+〈x, K⋆βz〉. Boundedness: ‖K⋆y‖^2 = |〈K⋆y, K⋆y〉| = |〈KK⋆y, y〉| ≤ ‖KK⋆y‖‖y‖ ≤ ‖K‖‖K⋆y‖‖y‖ ⇒ ‖K⋆y‖ ≤ ‖K‖ ‖y‖ ⇒ ‖K⋆| ≤ ‖K‖ < ∞.
(ii) 〈x, Ky〉 = 〈Ky, x〉 = 〈y, K⋆x〉 = 〈K⋆x, y〉 ⇒ 〈x, Ky〉 = 〈K⋆x, y〉 ∀x, y ∈ H. Replacing K by K⋆^ in Definition 1.1, we have 〈K⋆x, y〉 = 〈x, K⋆⋆y〉 ∀x, y ∈ H ⇒ 〈x, Ky〉 = 〈x, K⋆⋆y〉 ∀x, y ∈ H ⇒ K⋆⋆^ = K.
(iii) It is shown in part (i) that ‖K⋆| ≤ ‖K‖ < ∞. To prove ‖K⋆‖ = ‖K‖, it suffices to show that ‖K‖ ≤ ‖K⋆|, which is obtained by replacing K with K⋆^ in part (ii).
Definition 1.2. (Invariance) A subspace V of a vector space H is called invariant under an operator T ∈ BL(H, H) if the following condition holds:
T (V ) ⊆ V, i.e., ∀x ∈ V, it follows that T x ∈ V
Theorem 1.2. Let H be a Hilbert space and let T ∈ BL(H, H). A closed subspace V of H is invariant under the operator T if and only if V ⊥^ is T ⋆-invariant.
To show (Strong convergence) ⇒ (Weak convergence), we proceed as follows: If f ∈ W ⋆, then it follows from linearity and boundedness of the functional f that
klim→∞ T^ k^ =s^ T^ ⇒ |
f (T x) − f (T kx)
| = 0 ∀x ∈ V ∀f ∈ W ⋆^ ⇒ (^) klim→∞ T k^ =w T
We prove falsity of the converse by two counterexamples, one for each case. (Strong convergence) ; (Convergence in operator norm): Let us define a sequence of bounded linear operators T k^ : ℓ 2 → ℓ 2 ∀k ∈ N as:
T kx = { (^0) ︸ , 0 , (^0) ︷︷, · · · , (^0) ︸ f irst k terms
, ξk+1, ξk+2, · · · }, where x , {ξn : n ∈ N}
Clearly, T k^ is a bounded linear operator, i.e., T k^ ∈ BL(ℓ 2 , ℓ 2 ). Since x ∈ ℓ 2 , it follows that
klim→∞ ‖T^ kx‖ℓ^2 = 0^ ⇒^ klim→∞ ‖T^ k‖^ =s^0 ℓ^2 →ℓ^2 However, the induced norm limk→∞ sup‖x‖ℓ 2 =1 ‖T kx‖ℓ 2 = 1 as seen by choosing x = { (^0) ︸ , 0 , (^0) ︷︷, · · · , (^0) ︸ f irst k terms
, ξk+1, ξk+2, · · · } with ‖x‖ℓ 2 = 1 ⇒ limk→∞ ‖T k‖ 6 =u 0 ℓ 2 →ℓ 2.
Therefore, (Strong convergence) ; (Convergence in operator norm).
(Weak convergence) ; (Strong convergence): Let us define a sequence of bounded linear operators T k^ : ℓ 2 → ℓ 2 ∀k ∈ N as:
T kx = { (^0) ︸, 0 , (^0) ︷︷, · · · , (^0) ︸ f irst k terms
, ξ 1 , ξ 2 , · · · }, where x , {ξn : n ∈ N}
Clearly, T k^ is a bounded linear operator, i.e., T k^ ∈ BL(ℓ 2 , ℓ 2 ). Furthermore, in this Hilbert space setting, it follows from the Riesz-Frech`et Theorem that every f ∈ ℓ⋆ 2 can be represented as:
f (x) = 〈x, y〉ℓ 2 =
∑^ ∞
n=
ξnηn, where y = {ηk : k ∈ N}
It follows by Cauch-Schwarz inequality that
|f (T kx)|^2 = |〈T kx, y〉|^2 ≤
∑^ ∞
n=
|ξn|^2
∑^ ∞
m=k+
|ηm|^2 → 0 as k → ∞
However,
‖T kx‖ℓ 2 = ‖x‖ℓ 2 ∀k ∈ N ⇒ (^) klim→∞ ‖T kx‖ℓ 2 6 = 0 ∀x 6 = (^0) ℓ 2 ⇒ (^) klim→∞ T k^6 =s 0 ℓ 2 →ℓ 2
Therefore, (Weak convergence) ; (Strong convergence).
1.2 Convolution in Time-varying Systems
Let H = L 2 [α, β], where [α, β] ⊂ R. Let h : [α, β] × [α, β] → R be bounded in L 2 [α, β] × L 2 [α, β] in the following sense, i.e.,
‖h(t, • )‖^2 ,
∫ (^) β
α
dτ |h(t, τ )|^2 < ∞
‖h(•, τ )‖^2 ,
∫ (^) β α
dt |h(t, τ )|^2 < ∞
‖h‖^2 ,
∫ (^) β
α
dt
∫ (^) β
α
dτ |h(t, τ )|^2 < ∞
Let T ∈ BL(H, H) be defined as:
T x
(t) ,
∫ (^) t α dτ h(t, τ^ )x(τ^ )^ ∀x^ ∈^ H^ ∀t^ ∈^ [α, β]. Then, boundedness of T follows from the fact that, ∀x ∈ H,
‖T x‖^2 =
∫ (^) β α
dt |(T x)|^2 =
∫ (^) β α
dt
∫ (^) t α
dτ h(t, τ )x(τ )
2
∫ (^) β α
dt
∣〈h(t,^ • ), x(•)〉
∫ (^) β α
dt ‖h(t, • )‖^2 ‖˙x‖^2 =
∫ (^) β α
dt
∫ (^) β α
dτ |h(t, τ )|^2 < ∞
Next we find the adjoint operator T ⋆.
Proposition 1.3. (Adjoint operator T ⋆) The adjoint operator T ⋆^ is given as:
(T ⋆y)(t) (^) ︸︷︷︸= L 2 −sense
∫ (^) β t
dτ h(τ, t) y(τ ) ∀y ∈ H ∀t ∈ [α, β]
Proof. Following Fubuni’s theorem, the order of integration is exchanged as:
∀x, y ∈ H, 〈T x, y〉 =
∫ (^) β α
dt
(∫ (^) t
α
dτ h(t, τ ) x(τ )
y(t) =
∫ (^) β α
dτ x(τ )
∫ (^) β τ
dt h(t, τ ) y(t)
By exchanging τ and t in the last double integral, it follows that
〈T x, y〉 =
∫ (^) β
α
dt x(t)
∫ (^) β
t
dτ h(τ, t) y(τ ) = 〈x,
∫ (^) β
t
dτ h(τ, t) y(τ )〉
Since 〈T x, y〉 = 〈x, T ⋆y〉, we have
x,
T ⋆y −
∫ (^) β t dτ h(τ, t)^ y(τ^ )
= 0 ∀x, y ∈ H. Hence,
(T ⋆y)(t) (^) ︸︷︷︸= L 2 −sense
∫ (^) β t
dτ h(τ, t) y(τ ) ∀y ∈ H ∀t ∈ [α, β]
= A(t)Φ(t, to)xo + Φ(t, t)B(t)u(t) +
∫ (^) t to
dτ (^) ∂t∂ Φ(t, τ )B(τ )u(τ )
= A(t)Φ(t, to)xo + B(t)u(t) +
∫ (^) t
to
dτ A(t)Φ(t, τ )B(τ )u(τ )
= A(t)
[
Φ(t, to)xo +
∫ (^) t
to
dτ Φ(t, τ )B(τ )u(τ )
]
- B(t)u(t) = A(t)x(t) + B(t)u(t
and also x(to) = Φ(t 0 , to)xo +
∫ (^) to to dτ^ Φ(t, τ^ )B(τ^ )u(τ^ ) =^ xo.
Proposition 1.5. (Adjoint System) The adjoint of the finite-dimensional linear system dx dt(t )= A(t)x(t) is dz dt(t )= −AH^ (t)z(t), whose state transition matrices are Φ(t, to) and ΦH^ (to, t), respectively.
Proof. It follows from Proposition 1.4 that Φ(t, to)Φ(to, t) = Φ(t, t) = I. It follows, by taking the time derivatives on both sides, that
∂ ∂t
Φ(t, to)Φ(to, t)
= 0 ⇒ ∂Φ( ∂tt, t o)Φ(to, t) = −Φ(t, to) ∂Φ( ∂tto, t)
⇒ A(t)Φ(t, to) Φ(to, t) = −Φ(t, to) ∂Φ( ∂tto , t) ⇒ A(t) = −Φ(t, to) ∂Φ( ∂tto, t)
Taking Hermitian on both sides yields: ∂ ∂t
ΦH^ (to, t)
ΦH^ (t, to) = −AH^ (t)
Therefore, the state transition matrix of the system dz dt(t )= −AH^ (t)z(t) is ΦH^ (to, t).
Remark 1.3. Note that, in the adjoint system dz dt(t )= −AH^ (t)z(t), the negative sign on the right hand side implies that time is running backwards, i.e., t could be replaced by −t and the adjoint of the operator A is A⋆^ = AH^ for linear finite- dimensional systems.
1.3 Unitary Operators and Equivalent Inner Product Spaces
Analogous to the concepts of homeomorphism in topological spaces and isomor- phism in vector spaces, we introduce an analog of unitary equivalence in inner product spaces.
Definition 1.6. (Unitary operators) Let V and W be two inner product spaces over the same field. Then, V and W are said to be unitarily equivalent if there is an isomorphism ϕ : V → W of V onto W that preserves inner products, i.e., 〈ϕ(x), ϕ(y)〉W = 〈x, y〉V ∀x, y ∈ V. The operator ϕ is called unitary.
Theorem 1.3. Let V and W be two inner product spaces over the same field. A mapping ϕ is an isometric isomorphism of V onto W if and only if ϕ is unitary.
Proof. The only issue here is to show that if ‖ϕ(x)‖ = ‖x‖ ∀x ∈ V , then 〈ϕ(x), ϕ(y)〉W = 〈x, y〉V ∀x, y ∈ V. The proof follows by use of the polarization identity.
Remark 1.4. Let ϕ : Fn^ → Fn^ be a unitary operator. The n rows of ϕ form an orthonormal basis for Fn, and n columns of ϕ form an orthonormal basis for Fn.
1.4 Normal and Self-adjoint Operators
This subsection is devoted to normal and self-adjoint operators.
Definition 1.7. (Normal and self-adjoint operators) Let H be a Hilbert space, and let T ∈ BL(H, H). Then, T is said to be normal if T T ⋆^ = T ⋆T , i.e., if T commutes with its adjoint T ⋆; and T is said to be self-adjoint if T = T ⋆.
Remark 1.5. If T is self-adjoint, then T is normal and if T is unitary, then T is normal. However, self-adjoint operators may or may not be unitary and the converse is also true. While all orthogonal projections are self-adjoint, they are not unitary except for the trivial case of the identity operator I.
Theorem 1.4. The set of all self-adjoint operators on a Hilbert space H is closed in BL(H, H).
Proof. Let {Lk} be a sequence of self-adjoint operators in BL(H, H). Let ‖L − Lk‖ → 0 as k → ∞, where l ∈ BL(H, H). It suffices to show that L is self-adjoint, i.e., 〈Lx, y〉 = 〈x, Ly〉 ∀x, y ∈ H. Since Lk^ is self-adjoint for all k ∈ N, it follows that |〈Lx, y〉 − 〈x, Ly〉| = |〈Lx, y〉 − 〈Lkx, y〉| + 〈Lkx, y〉 − 〈x, Ly〉| = |〈(L − Lk)x, y〉 + 〈x, (L − Lk)y〉| ≤ 2 ‖L − Lk‖‖x‖‖y‖ → 0 as n → ∞.
Remark 1.6. It is not true, in general, that if A and B are self-adjoint, then AB is self-adjoint. However, if A and B are self-adjoint, then AB is self-adjoint if and only if AB = BA, i.e., if and only if A and B commute.
Remark 1.7. Let H be a Hilbert space, and let T ∈ BL(H, H). The following important results are presented below.
(i) If T is self-adjoint, then the operator norm of T is given by ‖T ‖ = sup‖x‖H =1〈T x, x〉 and ‖T ‖ = sup‖x‖H =1,‖y‖H =1〈T x, y〉.
(i) ∀ε > 0 there exists a finite-dimensional subspace M of the range space R(T ) such that infm∈M ‖T x − m‖W ≤ ε. In other words, the finite-dimensional space M is within the ε-radius of R(T ). Therefore, if R(T ) is infinite- dimensional, then smaller ε is, larger would be the dimension of M.
(ii) The following statements are equivalent.
- T is compact.
- If B is any bounded set in V , then the image T (B) lies in a compact subset of W.
- If B is any bounded set in V , then the image T (B) lies in a sequentially compact subset of W.
- If {xk} is a bounded sequence of vectors in V , then {T xk} contains a convergent subsequence in W.
- If B is any bounded set in V , then the image T (B) is a totally bounded set in W.
(iii) Let T ∈ L(V, W ) and s ∈ L(V, W ) be compact, where V and W are Banach spaces over the same field. Then, S + T is compact.
(iv) Let T ∈ L(V, V ) be compact, where V is a Banach space. Then, both ST and T S are compact.
(v) let {T k^ : V → W } be a sequence of compact linear transformations converging to T ∈ BL(V, W ), i.e., ‖T k^ − T ‖ → 0 as k → ∞. Then, T is compact. Hence the space of compact linear transformations is a closed subspace of bounded linear transformations.
