Metric and Hilbert Spaces

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 4 November 2014

Lecture 35: Linear operators on Hilbert spaces

(Riesz representation Theorem) Let $H$ be a Hilbert space and $H^{*}=B(H,ℂ)$ the dual of $H\text{.}$ Then $$\begin{array}{ccc}H& ⟶& H^{*}\\ x& ⟼& {\phi }_x:& H& ⟶& ℂ\\ & & & y& ⟼& ⟨y,x⟩\end{array}$$ is a bijective linear transformation with if $x\in X$ then $\Vert \phi x\Vert =\Vert x\Vert \text{.}$

HW: Let $H_1$ and $H_2$ be Hilbert spaces and let $T:H_1\to H_2$ be a bounded linear operator. Show that the adjoint of $T$ is $$T^{*}:H_2⟶H_1$$ given by if $x\in H_1$ and $y\in H_2$ then $${⟨Tx,y⟩}_1={⟨x,T^{*}y⟩}_2\text{.}$$

Let $H$ be a Hilbert space and let $T:H\to H$ be a bounded linear operator.

(a)	$T$ is self adjoint if $T=T^{*}\text{.}$
(b)	$T$ is positive if $T=T^{*}$ and if $x\in H$ then $⟨Tx,x⟩\in {ℝ}_{\ge 0}\text{.}$
(c)	$T$ is unitary if $T{T}^{*}=T^{*}T=I\text{.}$
(d)	$T$ is an isometry if $T$ satisfies if $x,y\in H$ then ${⟨Tx,Ty⟩}_2={⟨x,y⟩}_1\text{.}$

Let $X$ be a normed vector space. Let $$S=\{x\in X\hspace{0.17em}|\hspace{0.17em}\Vert x\Vert =1\}\text{.}$$ A bounded linear operator $T:X\to X$ is compact if $\overline{T\left(S\right)}$ is compact.

(13.3) Let $H$ be a Hilbert space. Let $T:H\to H$ be a bounded self adjoint operator. Then $$\Vert T\Vert =\text{sup}\left\{\mid ⟨Tx,x⟩\mid \hspace{0.17em}\right|\hspace{0.17em}\Vert x\Vert =1\}\text{.}$$

Let $H$ be a Hilbert space and let $T:H\to H$ be a nonzero self adjoint compact operator.

(a)	There exists $x\in H$ such that $\Vert x\Vert =1$ and if $u\in H$ and $\Vert u\Vert =1$ then $\mid ⟨Tu,u⟩\mid \le \mid ⟨Tx,x⟩\mid \text{.}$ Then $x$ is an eigenvector of $T$ with eigenvalue $\lambda$ such that $\Vert \lambda \Vert =\Vert T\Vert \text{.}$
(b)	There is an orthonormal basis of $H$ consisting of eigenvectors of $T\text{.}$
(c)	Let $\mathrm{\Lambda }$ be the set of eigenvalues of $T$ and let $P\left(\mu \right):H\to H$ be the orthogonal projection onto $X_{\mu },$ the subspace of eigenvectors with eigenvalue $\mu \text{.}$ Then $$Tx=\sum _{\mu \in \mathrm{\Lambda }}\mu P\left(\mu \right)x,\text{for}\hspace{0.17em}x\in H\text{.}$$

Examples of Hilbert spaces

${ℂ}^n$ with $⟨,⟩:{ℂ}^n\times {ℂ}^n\to ℂ$ given by $$⟨(x_1,x_2,\dots ,x_n),(y_1,y_2,\dots ,y_n)⟩=x_1\overline{y_1}+x_2\overline{y_2}+\cdots +x_n\overline{y_n}\text{.}$$

${\ell }^2$ with $⟨,⟩:{\ell }^2\times {\ell }^2\to ℂ$ given by $$⟨(x_1,x_2,\dots ),(y_1,y_2,\dots )⟩=x_1\overline{y_1}+x_2\overline{y_2}+\cdots =\sum _{i\in {\mathbb{Z}}_{>0}}{x}_i\overline{y_i}\text{.}$$

$$L^2[a,b]=\{f:[a,b]\to ℂ\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is a limit of step functions and}\hspace{0.17em}{\Vert f\Vert }_{L^2}<\infty \}$$ with inner product $⟨,⟩:L^2[a,b]\times L^2[a,b]\to ℂ$ given by $$⟨f,g⟩={\int }_a^{b}f\left(t\right)\overline{g\left(t\right)}dt\text{.}$$

Notes and References

These are a typed copy of Lecture 35 from a series of handwritten lecture notes for the class Metric and Hilbert Spaces given on September 25, 2014.

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