Dependencies

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

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 \( \mathcal{M} \).

Let \(X\) be a locally compact Hausdorff space. Associated to every bounded linear functional \(\Phi \) on \(C_0(X)\) we define a regular complex Borel measure \(\mu \) which we call the Riesz Measure associated to \(\Phi \).

TO DO: insert details from the proof of the exact definition.

Let \((X, \mathcal{A})\) be a measurable space and let \(Y\) be a Banach space. For a vector-valued measure \(\mu : \mathcal{A} \to Y\), the variation of \(\mu \) is the set function \(|\mu |: \mathcal{A} \to [0, +\infty ]\) defined by

for each \(E \in \mathcal{A}\).

If \(X\) is a locally compact Hausdorff space, then every bounded linear functional \(\Phi \) on \(C_0(X)\) is represented by a regular complex Borel measure \(\mu \), in the sense that

for every \(f \in C_0(X)\).

Consider a given bounded linear functional \(\Phi \) on \(C_0(X)\). Assume \(\| \Phi \| = 1\). (Update statement to be the general case.) We shall construct a positive linear functional \(\Lambda \) on \(C_c(X)\), such that

where \(\| f\| \) denotes the supremum norm.

For \(p \in \mathcal{B}(H)\) such that \(p = p^2 = p^*\), there is a closed subspace \(K\) such that \(p = p(K)\).

Moreover, the norm of \(\Phi \) is the total variation of \(\mu \):

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

It holds that \(p(K) = p(K)^2 = p(K)^*\).

For two \(\omega _1, \omega _2\), \( E(\omega _1), E(\omega _2) \) commute.

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 \(H\).

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

For any \(x \in H\), \( E_{x,x} \) is a positive measure on \( \mathfrak {M} \) whose total variation is:

For any \(x \in H\),

Let \(X\) be a locally compact Hausdorff space, and let \(\Phi \) be a bounded linear functional on \(C_0(X)\). Suppose that \(\mu \), \(\nu \) are regular complex Borel measure such that

Then \(\mu = \nu \).

Let \(\{ x_n\} \) be a sequence of pairwise orthogonal vectors in \(H\). Then the following are equivalent.

• \(\sum _{n=1}^\infty x_n\) converges in the norm topology of \(H\).

• \(\sum _{n=1}^\infty \| x_n\| ^2 {\lt} \infty \).

Let \(\{ x_n\} \) be a sequence of pairwise orthogonal vectors in \(H\). Then the following are equivalent.

• \(\sum _{n=1}^\infty x_n\) converges in the norm topology of \(H\).

• \(\sum _{n=1}^\infty \langle x, y\rangle \) converges for all \(y \in H\).

Suppose \(\mu \) is a positive measure on \(\mathfrak {M}\), \(g \in L^1(\mu )\), and

Then

For \(1 \leq p {\lt} \infty \), \(C_c(X)\) is dense in \(L^p(\mu )\).

Placeholder to combine the three results which make up The Reisz Theorem.

Suppose \(1 \leq p {\lt} \infty \), \(\mu \) is a \(\sigma \)-finite positive measure on \(X\), and \(\Phi \) is a bounded linear functional on \(L^p(\mu )\). Then there is a unique \(g \in L^q(\mu )\), where \(q\) is the exponent conjugate to \(p\), such that

Moreover, if \(\Phi \) and \(g\) are related as in (1), we have

In other words, \(L^q(\mu )\) is isometrically isomorphic to the dual space of \(L^p(\mu )\), under the stated conditions.

Let \(\mu \) be a complex measure on a \(\sigma \)-algebra \(\mathfrak {M}\) in \(X\). Then there is a measurable function \(h\) such that \(|h(x)| = 1\) for all \(x \in X\) and such that

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

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

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

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

If \(A\) is a closed normal subalgebra of \(\mathcal{B}(H)\) which contains the identity operator \(I\) and if \(\Delta \) is the maximal ideal space of \(A\), then the following assertions are true:

1. There exists a unique resolution \(E\) of the identity on the Borel subsets of \(\Delta \) which satisfies

   \begin{equation} \label{eq:1} T = \int _\Delta \widehat{T} \ dE \end{equation}

   1

   for every \(T \in A\), where \(\widehat{T}\) is the Gelfand transform of \(T\).

2. The inverse of the Gelfand transform (i.e., the map that takes \(\widehat{T}\) back to \(T\)) extends to an isometric *-isomorphism of the algebra \(L^\infty (E)\) onto a closed subalgebra \(B\) of \(\mathcal{B}(H)\), \(B\supset A\), given by

   \begin{equation} \label{eq:2} \Phi f = \int _\Delta f \ dE \quad (f \in L^\infty (E)). \end{equation}

   2

   Explicitly, \(\Phi \) is linear and multiplicative and satisfies

   \[ \Phi (\bar{f}) = (\Phi f)^*, \| \Phi f \| = \| f \| _{\infty } \quad (f \in L^\infty (E)). \]

3. \(B\) is the closure [in the norm topology of \(\mathcal{B}(H)\)] of the set of all finite linear combinations of the projections \(E(\omega )\).

4. If \(\omega \subset \Delta \) is open and nonempty, then \(E(\omega ) \neq 0\).

5. An operator \(S \in \mathcal{B}(H)\) commutes with every \(T \in A\) if and only if \(S\) com mutes with every projection \(E(\omega )\).