µ(X) = X n≥1 µn(σ(L)) = X n≥1 2−n<∞
So, (X, µ) is a finite measure space. As we shown in the first part of this section, we have unitary maps
Un;L2(σ(L), µn)→Hn
for each Hn such that Un−1LUn =Mx. Consider a surjective map V :L2(X, µ)→ L n≥1L2(σ(L), µn) mapping f tof(n, x) and: Z X f(x)dµ(x) = X n≥1 Z σ(L) f(n, x)dµn(x) We constructU by defining: U(X n≥1 vn) =V−1( X n≥1 Un−1(vn))
for all vn∈Hn. Since allHn span H, thus it is a unitary map defined on H with
inverse:
U−1(f) = X
n≥1
Un(f(n, x))
Finally, note that the n-th component of U(P
n≥1Un(f(n, x))) is f(n, x). There-
fore, the n-th component of U(L(P
n≥1Un(f(n, x)))) is
Un−1(LUnf(n, x)) = xf(n, x)
It implies that U ◦T = Mg ◦U where g is a bounded and measurable function
such that g = X →C (n, x)7→x
3.4
Spectral Theorem for Bounded Normal Op-
erators
Another way to interpret the spectral theorem is using resolutions of the identity (or projection-valued measures).
Definition 3.12. Let H be a Hilbert space and M be aσ-algebra over a set Ω. A resolution of the identity on M is a mapping:
E :M → B(H) with the following properties. For all ω, ω0, ω00∈M: (a) E(∅) = 0, E(Ω) =I.
(b) Each E(ω) is a self-adjoint projection (c) E(ω0∩ω00) = E(ω0)E(ω00)
(d) If ω0∩ω00 =∅, thenE(ω0∪ω00) = E(ω0) +E(ω00).
(e) For every x, y ∈H, the function E :M →C defined by: Ex,y(ω) =
E(ω)x, y is a complex measure.
Remark 3.13. Property (b) implies Ex,x(ω) =
E(ω)x, x = kE(ω)xk2 for all x ∈H. Each Ex,x is a positive measure with total variationkEx,xk=Ex.x(Ω) =
kxk2. Property (c) shows that any two of the projections commute. Properties (a) and (c) imply that E(ω0) andE(ω00) are orthogonal to each others ifω0∩ω00=∅.
There are some further important consequences.
Proposition 3.14. Consider a sequence of the disjoint sets (ωn)⊂M such that
ω =S n≥1ωn. For fixed x∈H, ∞ X n=1 E(ωn)x=E(ω)x
Proof. Properties (a) and (c) imply that {E(ωn)x}n≥1 is a sequence of pairwise orthogonal vectors in H. By property (e),
∞ X n=1 E(ωn)x, y =E(ω)x, y
for all y∈H. Sincexis fixed, thenP∞
n=1E(ωn) converges to E(ω) in the strong operator topology on B(H).
3.4. SPECTRAL THEOREM FOR BOUNDED NORMAL OPERATORS 43 For some functions, we can construct a bounded operator via resolutions of identity. Let f be a complex M-measurable function on Ω. Suppose that there exists a countable sequence of open discs {Di} such that E(f−1(Di)) = 0. Let
V be the union of Di. Thus, Ex,x(f−1(Di)) = 0 for all x ∈ H. By countably
additivity of the norm topology ofH,Ex,x(f−1(V)) = 0 and thenE(f−1(V)) = 0.
The complement of V is called the essential range of f. We can give a topology to the space of such f by defining the norm as the largest value through the essential range of f, which is denoted by k.k∞. Let B be the algebra of all bounded complex M-measurable functions on Ω with norm:
kfk= sup{|f(p)|:p∈Ω}
and a subset N ⊂B such that
N ={f ∈B :kfk∞ = 0}
Hence, we denote B/N byL∞(E). Indeed, it is a Banach algebra. Note that the Banach algebra is an associative algebra over the complex numbers and also a Banach space with a norm satisfying multiplicative inequality. The norm of any coset [f] is equal to kfk∞.
Now, we can construct a operator from resolutions of the identity.
Theorem 3.15. [18, Theorem 12.21 on p.319] LetH be a Hilbert space and let E
be a resolution of the identity. For any bounded Borel function f ∈L∞(E), there exists an isometric-isomorphism Φ : L∞(E) → A, given A is a closed normal subalgebra of B(H) such that:
Φ(f)x, y = Z Ω f dEx,y for x, y ∈H.
Remark 3.16. A normal subalgebra A of B(H) is a commutative subalgebra containing L∗ for every L∈ A. A map Φ is said to be isometric-isomorphism if Φ is linear, one to one and
Φ(f) = Φ(f)∗ for all f ∈L∞(E).
Now, we start to develop a spectral theorem for bounded normal operators in the resolutions of the identity form. We need to construct an approximate resolution of the identity, To do so, we use Gelfand transforms. (see [18, p.280])
Definition 3.17. Let ∆ be the set of all complex homomorphisms of a commu- tative Banach algebra A. Let x∈A xˆ: ∆ →C defined by
ˆ
x(h) = h(x) (h ∈∆) is called the Gelfand transform of x .
The Gelfand topology of ∆ is the weakest topology that makes every ˆx con- tinuous. In other words, it is the weak topology induced by ˆA, where ˆA is the set of all ˆx for x ∈ A. The set ∆ equipped with the Gelfand topology is called the maximal maximal ideal space of A.
Theorem 3.18. [18, Theorem 12.22 on p.321] IfA is a closed normal subalgebra of B(H) which contains the identity operator I and if ∆ is the maximal ideal space of A, then the following assertions are true:
(a) There exits a unique resolution E of the identity on the Borel subsets of ∆
which satisfies: L= Z ∆ ˆ LdE
for every L∈A, where Lˆ is the Gelfand transform of L.
(b) The inverse of the Gelfand transform extends to an isometric-isomorphism Φ
of the algebra L∞(E) onto a closed subalgebra B of B(H), A⊂B, given by
Φf =
Z
∆ f dE
Moreover, Φ is linear and multiplicative and satisfies:
Φf = (Φf)∗, kΦfk=kfk∞
for all f ∈L∞(E).
(c) The subalgebra B is the closure of the set of all finite linear combinations of the projections E(ω) in the topology norm of B(H).