The Spectral Theorem

4 Resolutions of the identity

  • Destination: ?

  • Principal reference: Chapter 12 of [Walter Rudin, Functional Analysis.][Rud87].

Definition 16 Rudin 12.17
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Let \(\mathfrak {M}\) be a \(\sigma \)-algebra in a set \(\Omega \), and let \(H\) be a Hilbert space. For simplicity, we assume that \(\Omega \) is a locally compact (Hausdorff) space. In this setting, a resolution of the identity (on \(\mathfrak {M}\)) is a mapping

\[ E: \mathfrak {M} \to \mathfrak {M}(H) \]

with the following properties:

  1. \( E(\emptyset ) = 0\), \(E(\Omega ) = I\).

  2. Each \( E(\omega ) \) is a self-adjoint projection.

  3. \( E(\omega ' \cap \omega '') = E(\omega ')E(\omega '')\).

  4. If \( \omega ' \cap \omega '' = \emptyset \), then \( E(\omega ' \cup \omega '') = E(\omega ') + E(\omega '') \).

  5. For every \( x \in H \) and \( y \in H \), the set function \( E_{x,y} \) defined by:

    \[ E_{x,y}(\omega ) = (E(\omega )x, y) \]

    is a complex regular Borel measure on \( \mathcal{M} \).

Lemma 17
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For any \(x \in H\),

\[ E_{x,x}(\omega ) = (E(\omega )x, x) = \| E(\omega )x\| ^2. \]
Proof ▶
Lemma 18
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For any \(x \in H\), \( E_{x,x} \) is a positive measure on \( \mathfrak {M} \) whose total variation is:

\[ \| E_{x,x}\| = E_{x,x}(\Omega ) = \| x\| ^2. \]
Proof ▶
Lemma 19
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For two \(\omega _1, \omega _2\), \( E(\omega _1), E(\omega _2) \) commute.

Proof ▶

By (3), any two of the projections \( E(\omega ) \) commute with each other.

Lemma 20
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If \( \omega ' \cap \omega '' = \emptyset \), then the ranges of \( E(\omega ') \) and \( E(\omega '') \) are orthogonal to each other

Proof ▶

By (1), (3) and Theorem 12.14.

Lemma 21
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If \(\{ \omega _j\} \) is a finite family of mutually disjoint Borel sets, then \(E(\bigcup _j \omega _j) = \sum _j E(\omega _j)\).

Proof ▶

By (4) and induction.

Remark: \(\sum _{n=1}^{\infty } E(\omega _n)\) does not converge in the norm topology of \(\mathcal{B}(H)\).

Lemma 22
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Let \(x \in H\) and \(\{ \omega _j\} \) be a countable family of mutually disjoint Borel sets. Then \(E(\bigcup _j \omega _j)x = \sum _j E(\omega _j)x\), where the right-hand side converges in the norm topology of \(H\).

Proof ▶

Since \( E(\omega _n)E(\omega _m) = 0 \) when \( n \neq m \), the vectors \( E(\omega _n)x \) and \( E(\omega _m)x \) are orthogonal to each other (Theorem 12.14). By (5),

\begin{equation} \label{eq:R5} \sum _{n=1}^{\infty } (E(\omega _n)x, y) = (E(\omega )x, y) \end{equation}
1

for every \( y \in H \). It now follows from Theorem 14 that:

\[ \sum _{n=1}^{\infty } E(\omega _n)x = E(\omega )x. \]

The series (1) converges in the norm topology of \( H \).

Proposition 23 Rudin 12.18
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If \( E \) is a resolution of the identity, and if \( x \in H \), then

\[ \omega \mapsto E(\omega )x \]

is a countably additive \( H \)-valued measure on* \( \mathfrak {M} \).

Proof ▶

This is the summary of what is proved above.

Moreover, sets of measure zero can be handled in the usual way:

Proposition 24 Rudin 12.19
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Suppose \( E \) is a resolution of the identity. If \( \omega _n \in \mathfrak {M} \) and \( E(\omega _n) = 0 \) for \( n = 1,2,3,\dots \), and if

\[ \omega = \bigcup _{n=1}^{\infty } \omega _n, \]

then \( E(\omega ) = 0 \).

Proof ▶

Since \( E(\omega _n) = 0 \), \( E_{x,x}(\omega _n) = 0 \) for every \( x \in H \). Since \( E_{x,x} \) is countably additive, it follows that \( E_{x,x}(\omega ) = 0 \). But

\[ \| E(\omega )x\| ^2 = E_{x,x}(\omega ). \]

Hence, \( E(\omega ) = 0 \).