Throughout, ${\displaystyle H}$  is a fixed Hilbert space.

A projection-valued measure on a measurable space ${\displaystyle (X,\Omega ),}$  where ${\displaystyle \Omega }$  is a σ-algebra of subsets of ${\displaystyle X,}$  is a mapping ${\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}$  such that for all ${\displaystyle \omega \in \Omega ,}$  ${\displaystyle \pi (\omega )}$  is a self-adjoint projection on ${\displaystyle H}$  (that is, ${\displaystyle \pi (\omega )}$  is a bounded linear operator ${\displaystyle \pi (\omega ):H\to H}$  that satisfies ${\displaystyle \pi (\omega )=\pi (\omega )^{*}}$  and ${\displaystyle \pi (\omega )\circ \pi (\omega )=\pi (\omega )}$ ) such that

A resolution of identity^[2] on a measurable space ${\displaystyle (X,\Omega )}$  is a function ${\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}$  such that for every ${\displaystyle \omega _{1},\omega _{2}\in \Omega }$ :

1. ${\displaystyle \pi (\varnothing )=0}$ ;
2. ${\displaystyle \pi (X)=\operatorname {Id} _{H}}$ ;
3. for every ${\displaystyle \omega \in \Omega ,}$  ${\displaystyle \pi (\omega )}$  is a self-adjoint projection on ${\displaystyle H}$ ;
4. for every ${\displaystyle x,y\in H,}$  the map ${\displaystyle \pi _{x,y}:\Omega \to \mathbb {C} }$  defined by ${\displaystyle \pi _{x,y}(\omega )=\langle \pi (\omega )x,y\rangle }$  is a complex measure on ${\displaystyle \Omega }$ ;
5. ${\displaystyle \pi \left(\omega _{1}\cap \omega _{2}\right)=\pi \left(\omega _{1}\right)\circ \pi \left(\omega _{2}\right)}$ ;
6. if ${\displaystyle \omega _{1}\cap \omega _{2}=\varnothing }$  then ${\displaystyle \pi \left(\omega _{1}\cup \omega _{2}\right)=\pi \left(\omega _{1}\right)+\pi \left(\omega _{2}\right)}$ ;

If ${\displaystyle \Omega }$  is the ${\displaystyle \sigma }$ -algebra of all Borels sets on a Hausdorff locally compact (or compact) space, then the following additional requirement is added:

7. for every ${\displaystyle x,y\in H,}$  the map ${\displaystyle \pi _{x,y}:\Omega \to \mathbb {C} }$  is a regular Borel measure (this is automatically satisfied on compact metric spaces).

Conditions 2, 3, and 4 imply that ${\displaystyle \pi }$  is a projection-valued measure.

Properties edit

Throughout, let ${\displaystyle \pi }$  be a resolution of identity. For all ${\displaystyle x\in H,}$  ${\displaystyle \pi _{x,x}:\Omega \to \mathbb {C} }$  is a positive measure on ${\displaystyle \Omega }$  with total variation ${\displaystyle \left\|\pi _{x,x}\right\|=\pi _{x,x}(X)=\|x\|^{2}}$  and that satisfies ${\displaystyle \pi _{x,x}(\omega )=\langle \pi (\omega )x,x\rangle =\|\pi (\omega )x\|^{2}}$  for all ${\displaystyle \omega \in \Omega .}$ ^[2]

For every ${\displaystyle \omega _{1},\omega _{2}\in \Omega }$ :

• ${\displaystyle \pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right)}$  (since both are equal to ${\displaystyle \pi \left(\omega _{1}\cap \omega _{2}\right)}$ ).^[2]
• If ${\displaystyle \omega _{1}\cap \omega _{2}=\varnothing }$  then the ranges of the maps ${\displaystyle \pi \left(\omega _{1}\right)}$  and ${\displaystyle \pi \left(\omega _{2}\right)}$  are orthogonal to each other and ${\displaystyle \pi \left(\omega _{1}\right)\pi \left(\omega _{2}\right)=0=\pi \left(\omega _{2}\right)\pi \left(\omega _{1}\right).}$ ^[2]
• ${\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}$  is finitely additive.^[2]
• If ${\displaystyle \omega _{1},\omega _{2},\ldots }$  are pairwise disjoint elements of ${\displaystyle \Omega }$  whose union is ${\displaystyle \omega }$  and if ${\displaystyle \pi \left(\omega _{i}\right)=0}$  for all ${\displaystyle i}$  then ${\displaystyle \pi (\omega )=0.}$ ^[2]
• However, ${\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}$  is countably additive only in trivial situations as is now described: suppose that ${\displaystyle \omega _{1},\omega _{2},\ldots }$  are pairwise disjoint elements of ${\displaystyle \Omega }$  whose union is ${\displaystyle \omega }$  and that the partial sums ${\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)}$  converge to ${\displaystyle \pi (\omega )}$  in ${\displaystyle {\mathcal {B}}(H)}$  (with its norm topology) as ${\displaystyle n\to \infty }$ ; then since the norm of any projection is either ${\displaystyle 0}$  or ${\displaystyle \geq 1,}$  the partial sums cannot form a Cauchy sequence unless all but finitely many of the ${\displaystyle \pi \left(\omega _{i}\right)}$  are ${\displaystyle 0.}$ ^[2]
• For any fixed ${\displaystyle x\in H,}$  the map ${\displaystyle \pi _{x}:\Omega \to H}$  defined by ${\displaystyle \pi _{x}(\omega ):=\pi (\omega )x}$  is a countably additive ${\displaystyle H}$ -valued measure on ${\displaystyle \Omega .}$ 
  • Here countably additive means that whenever ${\displaystyle \omega _{1},\omega _{2},\ldots }$  are pairwise disjoint elements of ${\displaystyle \Omega }$  whose union is ${\displaystyle \omega ,}$  then the partial sums ${\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)x}$  converge to ${\displaystyle \pi (\omega )x}$  in ${\displaystyle H.}$  Said more succinctly, ${\displaystyle \sum _{i=1}^{\infty }\pi \left(\omega _{i}\right)x=\pi (\omega )x.}$ ^[2]
  • In other words, for every pairwise disjoint family of elements ${\displaystyle \left(\omega _{i}\right)_{i=1}^{\infty }\subseteq \Omega }$  whose union is ${\displaystyle \omega _{\infty }\in \Omega }$ , then ${\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)=\pi \left(\bigcup _{i=1}^{n}\omega _{i}\right)}$  (by finite additivity of ${\displaystyle \pi }$ ) converges to ${\displaystyle \pi \left(\omega _{\infty }\right)}$  in the strong operator topology on ${\displaystyle {\mathcal {B}}(H)}$ : for every ${\displaystyle x\in H}$ , the sequence of elements ${\displaystyle \sum _{i=1}^{n}\pi \left(\omega _{i}\right)x}$  converges to ${\displaystyle \pi \left(\omega _{\infty }\right)x}$  in ${\displaystyle H}$  (with respect to the norm topology).

The ${\displaystyle \pi :\Omega \to {\mathcal {B}}(H)}$  be a resolution of identity on ${\displaystyle (X,\Omega ).}$ 

Essentially bounded functions edit

Suppose ${\displaystyle f:X\to \mathbb {C} }$  is a complex-valued ${\displaystyle \Omega }$ -measurable function. There exists a unique largest open subset ${\displaystyle V_{f}}$  of ${\displaystyle \mathbb {C} }$  (ordered under subset inclusion) such that ${\displaystyle \pi \left(f^{-1}\left(V_{f}\right)\right)=0.}$ ^[3] To see why, let ${\displaystyle D_{1},D_{2},\ldots }$  be a basis for ${\displaystyle \mathbb {C} }$ 's topology consisting of open disks and suppose that ${\displaystyle D_{i_{1}},D_{i_{2}},\ldots }$  is the subsequence (possibly finite) consisting of those sets such that ${\displaystyle \pi \left(f^{-1}\left(D_{i_{k}}\right)\right)=0}$ ; then ${\displaystyle D_{i_{1}}\cup D_{i_{2}}\cup \cdots =V_{f}.}$  Note that, in particular, if ${\displaystyle D}$  is an open subset of ${\displaystyle \mathbb {C} }$  such that ${\displaystyle D\cap \operatorname {Im} f=\varnothing }$  then ${\displaystyle \pi \left(f^{-1}(D)\right)=\pi (\varnothing )=0}$  so that ${\displaystyle D\subseteq V_{f}}$  (although there are other ways in which ${\displaystyle \pi \left(f^{-1}(D)\right)}$  may equal 0). Indeed, ${\displaystyle \mathbb {C} \setminus \operatorname {cl} (\operatorname {Im} f)\subseteq V_{f}.}$ 

The essential range of ${\displaystyle f}$  is defined to be the complement of ${\displaystyle V_{f}.}$  It is the smallest closed subset of ${\displaystyle \mathbb {C} }$  that contains ${\displaystyle f(x)}$  for almost all ${\displaystyle x\in X}$  (that is, for all ${\displaystyle x\in X}$  except for those in some set ${\displaystyle \omega \in \Omega }$  such that ${\displaystyle \pi (\omega )=0}$ ).^[3] The essential range is a closed subset of ${\displaystyle \mathbb {C} }$  so that if it is also a bounded subset of ${\displaystyle \mathbb {C} }$  then it is compact.

The function ${\displaystyle f}$  is essentially bounded if its essential range is bounded, in which case define its essential supremum, denoted by ${\displaystyle \|f\|^{\infty },}$  to be the supremum of all ${\displaystyle |\lambda |}$  as ${\displaystyle \lambda }$  ranges over the essential range of ${\displaystyle f.}$ ^[3]

Space of essentially bounded functions edit

Let ${\displaystyle {\mathcal {B}}(X,\Omega )}$  be the vector space of all bounded complex-valued ${\displaystyle \Omega }$ -measurable functions ${\displaystyle f:X\to \mathbb {C} ,}$  which becomes a Banach algebra when normed by ${\displaystyle \|f\|_{\infty }:=\sup _{x\in X}|f(x)|.}$  The function ${\displaystyle \|\,\cdot \,\|^{\infty }}$  is a seminorm on ${\displaystyle {\mathcal {B}}(X,\Omega ),}$  but not necessarily a norm. The kernel of this seminorm, ${\displaystyle N^{\infty }:=\left\{f\in {\mathcal {B}}(X,\Omega ):\|f\|^{\infty }=0\right\},}$  is a vector subspace of ${\displaystyle {\mathcal {B}}(X,\Omega )}$  that is a closed two-sided ideal of the Banach algebra ${\displaystyle \left({\mathcal {B}}(X,\Omega ),\|\cdot \|_{\infty }\right).}$ ^[3] Hence the quotient of ${\displaystyle {\mathcal {B}}(X,\Omega )}$  by ${\displaystyle N^{\infty }}$  is also a Banach algebra, denoted by ${\displaystyle L^{\infty }(\pi ):={\mathcal {B}}(X,\Omega )/N^{\infty }}$  where the norm of any element ${\displaystyle f+N^{\infty }\in L^{\infty }(\pi )}$  is equal to ${\displaystyle \|f\|^{\infty }}$  (since if ${\displaystyle f+N^{\infty }=g+N^{\infty }}$  then ${\displaystyle \|f\|^{\infty }=\|g\|^{\infty }}$ ) and this norm makes ${\displaystyle L^{\infty }(\pi )}$  into a Banach algebra. The spectrum of ${\displaystyle f+N^{\infty }}$  in ${\displaystyle L^{\infty }(\pi )}$  is the essential range of ${\displaystyle f.}$ ^[3] This article will follow the usual practice of writing ${\displaystyle f}$  rather than ${\displaystyle f+N^{\infty }}$  to represent elements of ${\displaystyle L^{\infty }(\pi ).}$ 

The maximal ideal space of a Banach algebra ${\displaystyle A}$  is the set of all complex homomorphisms ${\displaystyle A\to \mathbb {C} ,}$  which we'll denote by ${\displaystyle \sigma _{A}.}$  For every ${\displaystyle T}$  in ${\displaystyle A,}$  the Gelfand transform of ${\displaystyle T}$  is the map ${\displaystyle G(T):\sigma _{A}\to \mathbb {C} }$  defined by ${\displaystyle G(T)(h):=h(T).}$  ${\displaystyle \sigma _{A}}$  is given the weakest topology making every ${\displaystyle G(T):\sigma _{A}\to \mathbb {C} }$  continuous. With this topology, ${\displaystyle \sigma _{A}}$  is a compact Hausdorff space and every ${\displaystyle T}$  in ${\displaystyle A,}$  ${\displaystyle G(T)}$  belongs to ${\displaystyle C\left(\sigma _{A}\right),}$  which is the space of continuous complex-valued functions on ${\displaystyle \sigma _{A}.}$  The range of ${\displaystyle G(T)}$  is the spectrum ${\displaystyle \sigma (T)}$  and that the spectral radius is equal to ${\displaystyle \max \left\{|G(T)(h)|:h\in \sigma _{A}\right\},}$  which is ${\displaystyle \leq \|T\|.}$ ^[4]

The above result can be specialized to a single normal bounded operator.

• Projection-valued measure – Mathematical operator-value measure of interest in quantum mechanics and functional analysis
• Spectral theory of compact operators
• Spectral theorem – Result about when a matrix can be diagonalized

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