Spectral Theorem for Compact Hermitian Operators: Difference between revisions

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== Theorem ==
== Theorem ==


Let $H$ be a [[Definition:Hilbert Space|Hilbert space]].
Let $\HH$ be a [[Definition:Hilbert Space|Hilbert space]].


Let $T \in B_0 \left({H}\right)$ be a [[Definition:Compact Linear Operator|compact]], [[Definition:Self-Adjoint Operator|self-adjoint operator]].
Let $T \in \map {B_0} \HH$ be a [[Definition:Compact Linear Operator|compact]] [[Definition:Hermitian Operator|Hermitian operator]].




Then there exists a (possibly [[Definition:Finite Sequence|finite]]) [[Definition:Sequence|sequence]] $\left({\lambda_n}\right)$ of distinct nonzero [[Definition:Eigenvalue|eigenvalues]] of $T$ such that:
Then there exists a (possibly [[Definition:Finite Sequence|finite]]) [[Definition:Sequence|sequence]] $\sequence {\lambda_n}$ of distinct nonzero [[Definition:Eigenvalue|eigenvalues]] of $T$ such that:


* $P_n P_m = P_m P_n = 0$ if $n \ne m$
:$(1): \quad P_n P_m = P_m P_n = 0$ if $n \ne m$
* $\displaystyle \lim_{k \to \infty} \left\Vert{T - \sum_{n=1}^k \lambda_n P_n}\right\Vert = 0$, that is, $T = \displaystyle \sum_{n=1}^\infty \lambda_n P_n$
:$(2): \quad \ds \lim_{k \mathop \to \infty} \norm {T - \sum_{n \mathop = 1}^k \lambda_n P_n}$, that is, $T = \ds \sum_{n \mathop = 1}^\infty \lambda_n P_n$


where $P_n$ is the [[Definition:Orthogonal Projection|orthogonal projection]] onto the [[Definition:Eigenspace|eigenspace]] of $\lambda_n$, and $\left\Vert{\cdot}\right\Vert$ denotes the [[Definition:Norm on Bounded Linear Transformation|norm on bounded linear operators]].
where:
:$P_n$ is the [[Definition:Orthogonal Projection|orthogonal projection]] onto the [[Definition:Eigenspace|eigenspace]] of $\lambda_n$
:$\norm {\, \cdot \,}$ denotes the [[Definition:Norm on Bounded Linear Transformation|norm on bounded linear operators]].


{{refactor|Split corollaries to subpages}}
{{refactor|Split corollaries to subpages}}
=== Corollary 1 ===
=== Corollary 1 ===


There exists a (possibly [[Definition:Finite Sequence|finite]]) [[Definition:Sequence|sequence]] $\left({\mu_n}\right)$ of [[Definition:Real Number|real numbers]] and a [[Definition:Basis (Hilbert Space)|basis]] $E = \left({e_n}\right)$ for $\left({\operatorname{ker} T}\right)^\perp$ such that:
There exists a (possibly [[Definition:Finite Sequence|finite]]) [[Definition:Sequence|sequence]] $\sequence {\mu_n}$ of [[Definition:Real Number|real numbers]] and a [[Definition:Basis (Hilbert Space)|basis]] $E = \sequence {e_n}$ for $\paren {\ker T}^\perp$ such that:


:$\forall h \in H: Th = \displaystyle \sum_{n=1}^\infty \left\langle{h, e_n}\right\rangle_H \mu_n e_n$
:$\forall h \in H: T h = \ds \sum_{n \mathop = 1}^\infty \innerprod h {e_n}_\HH \mu_n e_n$




=== Corollary 2 ===
=== Corollary 2 ===


If $T$ has [[Definition:Zero Subspace|trivial]] [[Definition:Kernel (Abstract Algebra)|kernel]], then $H$ is [[Definition:Finite|finite]] [[Definition:Dimension (Hilbert Space)|dimensional]].
If $T$ has [[Definition:Zero Subspace|trivial]] [[Definition:Kernel (Abstract Algebra)|kernel]], then $\HH$ is [[Definition:Finite Dimensional|finite dimensional]].




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== Sources ==
== Sources ==


* {{BookReference|A Course in Functional Analysis|1990|John B. Conway|prev=Compact Self-Adjoint Operator has Countable Point Spectrum|next=Eigenspace for Normal Operator is Reducing Subspace}}: $II.5.1-5, 9$
* {{BookReference|A Course in Functional Analysis|1990|John B. Conway|prev = Compact Self-Adjoint Operator has Countable Point Spectrum|next = Eigenspace for Normal Operator is Reducing Subspace}}: $\text {II}.5.1-5, 9$


[[Category:Linear Transformations on Hilbert Spaces]]
[[Category:Linear Transformations on Hilbert Spaces]]

Revision as of 10:50, 25 July 2021

Theorem

Let $\HH$ be a Hilbert space.

Let $T \in \map {B_0} \HH$ be a compact Hermitian operator.


Then there exists a (possibly finite) sequence $\sequence {\lambda_n}$ of distinct nonzero eigenvalues of $T$ such that:

$(1): \quad P_n P_m = P_m P_n = 0$ if $n \ne m$
$(2): \quad \ds \lim_{k \mathop \to \infty} \norm {T - \sum_{n \mathop = 1}^k \lambda_n P_n}$, that is, $T = \ds \sum_{n \mathop = 1}^\infty \lambda_n P_n$

where:

$P_n$ is the orthogonal projection onto the eigenspace of $\lambda_n$
$\norm {\, \cdot \,}$ denotes the norm on bounded linear operators.


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Corollary 1

There exists a (possibly finite) sequence $\sequence {\mu_n}$ of real numbers and a basis $E = \sequence {e_n}$ for $\paren {\ker T}^\perp$ such that:

$\forall h \in H: T h = \ds \sum_{n \mathop = 1}^\infty \innerprod h {e_n}_\HH \mu_n e_n$


Corollary 2

If $T$ has trivial kernel, then $\HH$ is finite dimensional.


Proof


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Also see


Sources

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