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Chapter 6: Problem 9

Suppose that \((X, \mathscr{F})\) is a measure space and \(E: \mathscr{F} \rightarrow \mathscr{B}(\mathscr{H})\) is a spectral measure. Show that if \(f_{1}\) and \(f_{2}\) are bounded measurable functions on \(X\) with \(f_{1}(x)=\) \(f_{2}(x)\) for all \(x\) in some carrier \(C\) for \(E\), then $$ \int f_{1} d E=\int f_{2} d E . $$

Short Answer

Expert verified
The integrals are equal since the functions are the same on the carrier set, where the measure is effective, making unwanted regions irrelevant.

Step by step solution

01

Define the Integral Over a Spectral Measure

The integral \( \int f dE \) with respect to a spectral measure \( E \) is defined using the spectral theorem. For a bounded measurable function \( f \), this integral is calculated in terms of the operator \( T = \int f dE \), which acts on a Hilbert space \( \mathscr{H} \). This operator is constructed through the limit of simple functions that approximate \( f \).
02

Understand the Concept of a Carrier

A carrier \( C \) for the spectral measure \( E \) is a measurable set in \( X \) such that \( E(X \setminus C) = 0 \). This implies that the measure outside \( C \) is zero, and it is sufficient to consider the function values inside \( C \).
03

Consider the Equality on the Carrier

Given that \( f_1(x) = f_2(x) \) for all \( x \in C \), we recognize that the functions \( f_1 \) and \( f_2 \) are identical on the set where \( E \) is relevant, since \( E(X \setminus C) = 0 \). Thus, they contribute identically to the integral with respect to \( E \).
04

Use the Property of the Integral with Respect to Spectral Measures

Using properties of integrals with respect to spectral measures, we know \( \int f_1 dE - \int f_2 dE \) primarily depends on regions where \( E eq 0 \). Since \( f_1 \) and \( f_2 \) coincide on \( C \) and \( E \) is zero on the complement, the property of the spectral measure implies that the integrals must be equal.
05

Conclude the Equality of the Integrals

Since the functions are equal on the carrier and the measure outside the carrier is irrelevant, we conclude \( \int f_{1} d E = \int f_{2} d E \). This equality stems from the fact that the integrands are equal where the measure contributes to the integration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Measure Space
A measure space is a foundational concept in measure theory, comprising a set equipped with a sigma-algebra and a measure. Specifically, a measure space is given by a triple \(X, \mathscr{F}, \mu\), where:\
    \
  • \(X\) is a set, often interpreted as a space where events occur or elements reside.
    • \
  • \(\mathscr{F}\) is a sigma-algebra on \(X\), a collection of subsets of \(X\) including the empty set, closed under complementation and countable unions.
    • \
  • \(\mu\) is a measure, a function assigning non-negative real numbers or infinity to the sets in \(\mathscr{F}\), respecting the countable additivity property.
    • \
\These components allow for rigorous integration and probability discussions by providing a context in which we can evaluate the size or probability of certain sets.
Integral Over a Spectral Measure
The integral over a spectral measure is an advanced concept in functional analysis, particularly within the framework of quantum mechanics and Hilbert spaces. A spectral measure \(E\) maps subsets of a sigma-algebra \(\mathscr{F}\) to a set of bounded linear operators \(\mathscr{B}(\mathscr{H}) \) on a Hilbert space \(\mathscr{H}\). This mapping allows us to generalize the concept of integration from measurable functions to operators:\
    \
  • For a bounded measurable function \(f\), the integral \(\int f \ dE \) is understood through the spectral theorem.
    • \
  • The process constructs an operator from simple functions that approximate \(f\), analogous to how definite integrals approximate cumulative distribution functions.
    • \
\This operator-valued integration is crucial for defining functional calculus of self-adjoint operators, enabling the translation of spectral properties of linear operators to observable quantities.
Carrier Set
The carrier set of a spectral measure is a significant yet intuitive concept. A spectral measure assigns zero outside its carrier, focusing our attention on where the action happens. For a spectral measure \(E\) defined on a measure space \(X\), a carrier \(C\) is a subset where essentially all contributions to any operation involving \(E\) occur:\
    \
  • The measure outside this subset is \( E(X \setminus C) = 0 \).
    • \
  • This ensures that operations, like integration over \(E\), only need to consider what happens within \(C\).
    • \
\The identification of a carrier simplifies many calculations, as it allows us to ignore any values of functions outside \(C\), knowing they do not affect the results of integrals or other measures.
Bounded Measurable Functions
Bounded measurable functions play an essential role when dealing with integration over measure spaces, ensuring certain mathematical processes converge nicely. A function is considered bounded on a measure space if there exists a real number \(M\) such that for all values in its domain, the absolute value of the function does not exceed \(M\):\
    \
  • \(f\) is measurable if, for every real number, the pre-image of all intervals of the form \(( -\infty, a ]\) is within the sigma-algebra \(\mathscr{F}\).
    • \
  • Functions satisfying these conditions are integral over spectral measures, as they ensure that approximations converge uniformly.
    • \
\These properties make bounded measurable functions particularly suitable for integration across diverse mathematical settings, ensuring well-defined results.

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Most popular questions from this chapter

A conjugation on a Hilbert space \(\mathscr{H}\) is a conjugate linear map \(C: \mathscr{H} \rightarrow \mathscr{H}\) with \(C^{2}=I\) and \(\langle C x, C y\rangle=\langle y, x\rangle\) for all \(x, y\) in \(\mathscr{H}\). (a) Show that if we fix an orthonormal basis \(\left\\{e_{n}\right\\}\) for \(\mathscr{H}\) and set $$ C\left(\sum \lambda_{n} e_{n}\right)=\sum \overline{\lambda_{n}} e_{n} $$ (provided \(\left\\{\lambda_{n}\right\\} \in \ell^{2}\) ) then \(C\) is a conjugation. (b) If \(\mathscr{H}=L^{2}(X, \mu)\) for some \(\sigma\)-finite measure space \((X, \mu)\), show that \(C(f)=\bar{f}\) is a conjugation. (c) If \(C\) is a conjugation, an operator \(T \in \mathscr{B}(\mathscr{H})\) is called \(C\)-symmetric if \(C T^{*}=\) \(T C\). Show that the Volterra operator \(V f(x)=\int_{0}^{x} f(t) d t\) on \(L^{2}[0,1]\) is \(C\)-symmetric for \(C f(x)=\overline{f(1-x)}\). (d) If \(N\) is any normal operator in \(\mathscr{B}(\mathscr{H})\), find a conjugation \(C\) on \(\mathscr{H}\) so that \(C N^{*}=\) \(N C\).

Suppose that \(\mathscr{H}\) is an infinite-dimensional separable Hilbert space and that \(T \in \mathscr{B}(\mathscr{H})\) is a compact normal operator. Let \(f\) be any function in \(C(\sigma(T))\). Show that \(f(T)\) is compact if and only if \(f(0)=0\).

Suppose \(E\) is a spectral measure and suppose that \(E(S)\) commutes with an operator \(T\) for each \(S\) in \(\mathscr{F}\). Show that for any \(f \in B(X, \mathscr{F}), \int f d E\) commutes with \(T\).

Let \(A\) be the normal operator of multiplication by \(\varphi(x)=x\) on \(L^{2}[0,1]\). Let \(E:[0,1] \rightarrow \mathscr{B}\left(L^{2}[0,1]\right)\) be the spectral measure of \(A\) as in Theorem 6.23. Identify \(E\).

Show that every normal operator \(A\) acting on a separable Hilbert space is unitarily equivalent to a multiplication operator on \(L^{2}(X, \mu)\) for some finite measure space \((X, \mu)\). Hint: If \(\mathscr{H}_{n}\) is invariant under \(C^{*}(A)\) and has cyclic vector \(h_{n}\), then we may assume \(\left\|h_{n}\right\|=2^{-n}\).
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