Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

Contents

Frontmatter

I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

Exercise IV.44

Let $$$X= \mathopen{}\left\{\, f\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}\,\middle\vert\, , f\mathopen{}\left( 1\right)\mathclose{}= 0, \,\right\}\mathclose{}$$$, made into a Banach space by endowing it with the uniform norm: $$$\mathopen{}\left\lVert{}f\right\rVert_\infty\mathclose{}= \max{} \mathopen{}\left\{\, \mathopen{}\left\lvert{}f\mathopen{}\left( t\right)\mathclose{}\right\rvert\mathclose{}\,\middle\vert\, , 0\leq t\leq 1, \,\right\}\mathclose{}$$$. Consider $$E= \mathopen{}\left\{\, g\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}\,\middle\vert\, , g\mathopen{}\left( 1\right)\mathclose{}= 0, , \int _{0}^{1}{}tg\mathopen{}\left( t\right)\mathclose{}\,\mathrm{d}t= 1, \,\right\}\mathclose{}\text{.}$$

(a)  Show that $$$E$$$ is a closed convex subset of $$$X$$$.

(b)  Find the distance from $$$0$$$ to $$$E$$$; that is, find $$$\inf_{g\in E}{}\mathopen{}\left\lVert{}g\right\rVert\mathclose{}_{\infty}$$$.

(c)  Show that there is no nearest point in $$$E$$$ to $$$0$$$; that is, show that there is no $$$g$$$ in $$$E$$$ for which $$$\mathopen{}\left\lVert{}g\right\rVert\mathclose{}_{\infty}= \inf_{g\in E}{}\mathopen{}\left\lVert{}g\right\rVert\mathclose{}_{\infty}$$$.

Exercise IV.45

Let $$$X$$$ be a separable normed linear space and let $$$\mathopen{}\left({x}_{i}\right)\mathclose{}_{i=1}^{\infty}$$$ be a sequence whose closure is the unit ball of $$$X$$$. For $$$f$$$ and $$$g$$$ in $$$X^{*}$$$, define $$$\operatorname{d}\mathopen{}\left( f, g\right)\mathclose{}= \sum{}{2}^{{-}n}\mathopen{}\left\lvert{}f\mathopen{}\left( {x}_{n}\right)\mathclose{}-g\mathopen{}\left( {x}_{n}\right)\mathclose{}\right\rvert\mathclose{}$$$. Show that $$$\operatorname{d}$$$ is a metric on $$$X^{*}$$$ and that $$$\operatorname{d}$$$ gives the $$$\mathop{\mathrm{w}^*}$$$ topology on the unit ball of $$$X^{*}$$$.

Exercise IV.46

Show that a bounded linear operator on a Hilbert space is compact if and only if it takes bounded weakly convergent nets to norm-convergent nets.

Exercise IV.47

Let $$$T$$$ be the Volterra operator on $$$H= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$, so $$$\mathopen{}\left(T\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= \int _{0}^{x}{}f\mathopen{}\left( t\right)\mathclose{}\,\mathrm{d}t$$$.

(a)  Show that $$$T$$$ has no eigenvalues.

(b)  Show that $$$\mathop{\sigma}\mathopen{}\left( T\right)\mathclose{}= \mathopen{}\left\{\, 0\,\right\}\mathclose{}$$$.

(c)  Prove the integration by parts formula $$$T\mathopen{}\left( ψ'T\mathopen{}\left( h\right)\mathclose{}\right)\mathclose{}= ψT\mathopen{}\left( h\right)\mathclose{}-T\mathopen{}\left( ψh\right)\mathclose{}$$$ for $$$ψ\in \mathrm{C}^{1}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$ and $$$h\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$. Advice: $$$\mathrm{C}^{1}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$ is dense in $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$.

(d)  Define $$${φ}_{λ}$$$ for non-zero $$$λ$$$ by $$${φ}_{λ}\mathopen{}\left( x\right)\mathclose{}= {\mathrm{e}}^{\frac{{-}x}{λ}}$$$. Show that $$${ \mathopen{}\left(λ-T\right)\mathclose{} }^{-1}$$$ is given on $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$ by the formula $${ \mathopen{}\left(λ-T\right)\mathclose{} }^{-1}\mathopen{}\left( f\right)\mathclose{}= \frac{1}{λ}f+\frac{1}{{λ}^{2}}\frac{1}{{φ}_{λ}}T\mathopen{}\left( {φ}_{λ}f\right)\mathclose{}\text{.}$$

Exercise IV.48

Let $$$T$$$ be the Volterra operator.

(a)  Find an orthonormal basis for $$$H$$$ consisting of eigenvectors for $$$T^{*}T$$$.

(b)  Find $$$\mathopen{}\left\lVert{}T\right\rVert\mathclose{}$$$.

(c)  Calculate $$$\operatorname{Tr}\mathopen{}\left( T^{*}T\right)\mathclose{}$$$ by summing a series. Use $$$\sum_{k=1}^{\infty}{}{k}^{{-}2}= \frac{{\mathrm{\pi}}^{2}}{6}$$$.

(d)  Write $$$T^{*}T$$$ as an integral operator on $$$H$$$ with a continuous kernel, and then use Theorem III.42 to calculate $$$\operatorname{Tr}\mathopen{}\left( T^{*}T\right)\mathclose{}$$$.

(e)  Write a series for $$$\mathopen{}\left\lvert{}T\right\rvert\mathclose{}$$$ that converges in the norm of $$$H$$$ at each $$$f\in H$$$. Express $$$\mathopen{}\left\lvert{}T\right\rvert\mathclose{}$$$ of the constant function $$$\mathbf{1}$$$, i.e. $$$\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathbf{1}$$$, as the sum of a uniformly convergent trigonometric series. (As a partial check, you should get $$$\mathopen{}\left(\mathopen{}\left\lvert{}T\right\rvert\mathclose{}\mathbf{1}\right)\mathclose{}\mathopen{}\left( 0\right)\mathclose{}= \frac{8}{{\mathrm{\pi}}^{2}}\sum_{n=0}^{\infty}{} \frac{{{-}1}^{n}}{{\mathopen{}\left(2n+1\right)\mathclose{}}^{2}} = \frac{8}{{\mathrm{\pi}}^{2}}G$$$, where $$$G$$$ is Catalan's constant.)

(f)  For $$$f\in H$$$ and $$$α\in \mathbb{C}\setminus \mathop{\sigma}\mathopen{}\left( T^{*}T\right)\mathclose{}$$$, let $$$g= { \mathopen{}\left(1-α T^{*}T\right)\mathclose{} }^{-1}f$$$. (i) Write $$$g$$$ as the sum of a trigonometric series whose coefficients depend on $$$α$$$ and $$$f$$$. (ii) Verify that $$$g$$$ is given by $$g\mathopen{}\left( t\right)\mathclose{}= f\mathopen{}\left( t\right)\mathclose{}+\sqrt{α}\mathopen{}\left(\frac{\cos\mathopen{}\left( \sqrt{α}t\right)\mathclose{}}{\cos\mathopen{}\left( \sqrt{α}\right)\mathclose{}}\int _{0}^{1}{} \sin\mathopen{}\left( \sqrt{α}\mathopen{}\left(1-θ\right)\mathclose{}\right)\mathclose{}f\mathopen{}\left( θ\right)\mathclose{} \,\mathrm{d}θ-\int _{0}^{t}{} \sin\mathopen{}\left( \sqrt{α}\mathopen{}\left(t-θ\right)\mathclose{}\right)\mathclose{}f\mathopen{}\left( θ\right)\mathclose{} \,\mathrm{d}θ\right)\mathclose{}\text{.}$$

Advice for (a), (b), (d): This is like the example in Section G, with $$$T^{*}T$$$ in place of $$$K$$$ and the differential operator $$$f\mapsto {-}f''$$$ with boundary conditions $$$f' \mathopen{}\left( 0\right)\mathclose{}= 0= f\mathopen{}\left( 1\right)\mathclose{}$$$ in place of $$$L$$$.

Exercise IV.49

(a)  Let $$$R$$$ be a ring with identity, and let $$$J$$$ be a proper ideal of $$$R$$$. Show that if $$$x\in J$$$ and $$$1-x$$$ is invertible, then $$$1-{\mathopen{}\left(1-x\right)\mathclose{}}^{-1}$$$ is invertible. Deduce that if the Hilbert space operator $$$K$$$ is compact (resp. Hilbert-Schmidt, trace class, finite-dimensional) and $$$1\notin \mathop{\sigma}\mathopen{}\left( K\right)\mathclose{}$$$, then $$${\mathopen{}\left(1-K\right)\mathclose{}}^{-1}-1$$$ is compact (resp. Hilbert-Schmidt, trace class, finite-dimensional).

(b)   In the situation of part (f) of Exercise IV.48, calculate the trace of $$${ \mathopen{}\left(1-α T^{*}T\right)\mathclose{} }^{-1}-1$$$ as the sum of a series using (i), and in closed form using (ii). (As a check, have Mathematica or the like sum the series exactly.)

Exercise IV.50

Let $$$S\in \mathcal{L}\mathopen{}\left( H\right)\mathclose{}$$$.

(a)  Show that if $$$η\in H$$$ satisfies $$$\mathopen{}\left\lvert{}\mathopen{}\left\langle{}ξ, η\right\rangle\mathclose{}\right\rvert\mathclose{}\leq M\mathopen{}\left\lVert{}Sξ\right\rVert\mathclose{}$$$ for some positive constant $$$M$$$ and all $$$ξ\in H$$$, then $$$η\in S^{*}H$$$. (Advice: That $$$η\in \overline{ S^{*}H }$$$ is easy. If you can get $$$η= \lim_{n}{} S^{*}{ρ}_{n}$$$ for a bounded sequence $$$\mathopen{}\left( {ρ}_{n} \right)\mathclose{}$$$, Alaoglu's Theorem will then close the sale.)

(b)  Suppose that $$$\operatorname{Ker}\mathopen{}\left( S\right)\mathclose{}= \mathopen{}\left\{\, 0\,\right\}\mathclose{}= \operatorname{Ker}\mathopen{}\left( S^{*}\right)\mathclose{}$$$. Define $$$L$$$ on the dense subspace $$$SH$$$ by $$$LSξ= ξ$$$. Show that $$$L^{*}$$$ has domain $$$S^{*}H$$$, and is given there by $$$L^{*} S^{*}ξ= ξ$$$. (In particular, the inverse of a bounded self-adjoint operator with kernel $$$\mathopen{}\left\{\, 0\,\right\}\mathclose{}$$$ is self-adjoint on its natural domain.)

Exercise IV.51

Your mission is to prove the following version of the spectral theorem.

Theorem S: Given a densely defined self-adjoint operator $$$h$$$ in the Hilbert space $$$H$$$, there exist a measure space $$$\mathopen{}\left(X, M, μ\right)\mathclose{}$$$, a measurable function $$$α : X \to \mathopen{}\left({-}\infty, \infty\right)\mathclose{}$$$, and a unitary map $$$U : H \to \mathrm{L}^{\mathrm{2}}\mathopen{}\left( XMμ\right)\mathclose{}$$$ such that $$$U\mathopen{}\left( \mathop{\mathcal{D}}\mathopen{}\left( h\right)\mathclose{}\right)\mathclose{}= \mathopen{}\left\{\, ξ\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( X\right)\mathclose{}\,\middle\vert\, , α\mathopen{}\left( ξ\right)\mathclose{}\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( X\right)\mathclose{}, \,\right\}\mathclose{}$$$ and $$$U\mathopen{}\left( h\mathopen{}\left( U^{*}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\right)\mathclose{}= αξ$$$ for all $$$ξ\in U\mathopen{}\left( \mathop{\mathcal{D}}\mathopen{}\left( h\right)\mathclose{}\right)\mathclose{}$$$.

In other words, $$$H$$$ is unitarily equivalent to multiplication by a measurable real function in an $$$\mathrm{L}^{\mathrm{2}}$$$-space.

The proof envisioned here requires an important result from measure theory at a level somewhat beyond the assumed prerequisite knowledge for this course. One of several theorems called the Riesz representation theorem says that if $$$Ω$$$ is a compact Hausdorff space and $$$Φ : \mathrm{C}\mathopen{}\left( Ω\right)\mathclose{} \to \mathbb{C}$$$ is a positive linear functional (i.e. $$$f\geq 0$$$ implies $$$Φ\mathopen{}\left( f\right)\mathclose{}\geq 0$$$, then there is a $$$σ$$$-algebra $$$M$$$ of subsets of $$$Ω$$$ and a measure $$$μ$$$ defined on $$$Ω$$$ such that (i) $$$M$$$ contains every open subset of $$$Ω$$$ (so every function in $$$\mathrm{C}\mathopen{}\left( Ω\right)\mathclose{}$$$ is measurable); (ii) $$$\mathrm{C}\mathopen{}\left( Ω\right)\mathclose{}$$$ is dense in $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( Ω M μ \right)\mathclose{}$$$; and (iii) $$$Φ\mathopen{}\left( f\right)\mathclose{}= \int _{Ω}{}f\,\mathrm{d}μ$$$ for all $$$f\in \mathrm{C}\mathopen{}\left( Ω\right)\mathclose{}$$$.

Start with the case of a bounded self-adjoint operator $$$A$$$ on $$$H$$$. For a vector $$$ξ\in H$$$, call the closure of the linear span of $$$\mathopen{}\left\{\, {A}^{n}\mathopen{}\left( ξ\right)\mathclose{}\,\middle\vert\, , n\geq 0, , n\in \mathbb{N}, \,\right\}\mathclose{}$$$ the cyclic subspace generated by $$$A$$$. If this subspace is all of $$$H$$$, i.e. $$$\overline{ \mathopen{}\left\{\, f\mathopen{}\left( A\right)\mathclose{}ξ\,\middle\vert\, , f\in \mathrm{C}\mathopen{}\left( \mathop{\sigma}\mathopen{}\left( A\right)\mathclose{}\right)\mathclose{}, \,\right\}\mathclose{} }= H$$$, we say that $$$ξ$$$ is a cyclic vector for $$$A$$$.

(a)  Prove Theorem S in the case of a bounded self-adjoint operator $$$A$$$ on $$$H$$$ with a cyclic vector. (Advice: $$$Ω= σ\mathopen{}\left( A\right)\mathclose{}$$$, $$$Φ\mathopen{}\left( f\right)\mathclose{}= \mathopen{}\left\langle{}f\mathopen{}\left( A\right)\mathclose{}ξ, ξ\right\rangle\mathclose{}$$$, and $$${\mathopen{}\left\lVert{}f\mathopen{}\left( A\right)\mathclose{}ξ\right\rVert\mathclose{}}^{2}= Φ\mathopen{}\left( {\mathopen{}\left\lVert{}f\right\rVert\mathclose{}}^{2}\right)\mathclose{}= {\mathopen{}\left(\mathopen{}\left\lVert{}f\right\rVert\mathclose{}_{ \mathrm{L}^{\mathrm{2}}\mathopen{}\left( Ωμ\right)\mathclose{} }\right)\mathclose{}}^{2}$$$.)

(b)  Show that in general, $$$H$$$ is the direct sum of mutually orthogonal cyclic subspaces for $$$A$$$.

(c)  Prove Theorem S for any bounded self-adjoint $$$A$$$. (Hint: The direct sum of measure spaces $$$\mathopen{}\left({X}_{i}, {M}_{i}, {μ}_{i}\right)\mathclose{}$$$ is the disjoint union of the $$${X}_{i}$$$, with the obvious measure concocted from the $$${μ}_{i}$$$ on the obvious $$$σ$$$-algebra determined by the $$${M}_{i}$$$.)

Now let $$$h$$$ be a self-adjoint operator in $$$H$$$. Form the bounded operator $$$A$$$ on $$$H$$$ as in Section A, that is, $$$A= h{\mathopen{}\left(1+{h}^{2}\right)\mathclose{}}^{{-}\frac{1}{2}}$$$, which as we have seen entails $$$A= A^{*}$$$, $$${-}1\leq A\leq 1$$$, $$$h= A{\mathopen{}\left(1-{A}^{2}\right)\mathclose{}}^{\frac{1}{2}}$$$, and $$$\operatorname{Ker}\mathopen{}\left( 1-{A}^{2}\right)\mathclose{}= \mathopen{}\left\{\, 0\,\right\}\mathclose{}$$$. Invoke (c) for $$$A$$$ to get $$$\mathopen{}\left(X, M, μ\right)\mathclose{}$$$, $$$α$$$, and $$$U$$$ doing what they're supposed to do. Move everything over to $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( X, μ\right)\mathclose{}$$$. Relabel $$$UhU$$$ as $$$h$$$, now acting on $$$H= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( X, μ\right)\mathclose{}$$$, and let $$$A$$$ be multiplication by $$$α$$$ on the new $$$H$$$. Notice that $$${-}1\lt α\lt 1$$$ (after redefining on a set of measure zero, if one is fussy). Consider now the measurable function $$$β$$$ defined by $$$β= α{\mathopen{}\left(1-{α}^{2}\right)\mathclose{}}^{{-}\frac{1}{2}} : X \to \mathopen{}\left({-}\infty, \infty\right)\mathclose{}$$$. Let $$$\mathrm{M}_{β}$$$ be multiplication by $$$β$$$, with domain $$$\mathop{\mathcal{D}}\mathopen{}\left( \mathrm{M}_{β}\right)\mathclose{}= \mathopen{}\left\{\, ξ\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( Xμ\right)\mathclose{}\,\middle\vert\, , βξ\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( Xμ\right)\mathclose{}, \,\right\}\mathclose{}$$$. It is useful at this point to introduce the subspaces $$${H}_{τ}= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( {β}^{-1}\mathopen{}\left( \mathopen{}\left[{-}τ, τ\right]\mathclose{}\right)\mathclose{}\right)\mathclose{}$$$ for $$$τ\gt 0$$$.

(d)   Show that $$$\mathrm{M}_{β}$$$ is self-adjoint. (Hint: Modify certain of the arguments in Section A.)

(e)   Show that $$$\mathop{\mathcal{G}}\mathopen{}\left( \mathrm{M}_{β}\right)\mathclose{}= \mathop{\mathcal{G}}\mathopen{}\left( h\right)\mathclose{}$$$, thus finishing the proof of Theorem S. (Same hint.)

Exercise IV.52

For compactly supported continuous functions $$$f : {\mathbb{R}}^{2} \to \mathbb{C}$$$, define $$${S}_{f}$$$ on $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}\right)\mathclose{}$$$ by $$$\mathopen{}\left({S}_{f}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( t\right)\mathclose{}= \int _{\mathbb{R}}{} f\mathopen{}\left( t, t-s\right)\mathclose{}ξ\mathopen{}\left( s\right)\mathclose{} \,\mathrm{d}s$$$.

(a)  Show that $$${S}_{f}\in \mathcal{L}\mathopen{}\left( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}\right)\mathclose{}\right)\mathclose{}$$$, with $$$\mathopen{}\left\lVert{}{S}_{f}\right\rVert\mathclose{}\leq \mathopen{}\left\lVert{}f\right\rVert\mathclose{}_{\mathrm{L}^{\mathrm{2}}\mathopen{}\left( {\mathbb{R}}^{2}\right)\mathclose{}}$$$.

(b)  Let $$$K= \mathopen{}\left\{\, {S}_{f}\,\middle\vert\, , f\in {\mathrm{C}}_{0}\mathopen{}\left( {\mathbb{R}}^{2}\right)\mathclose{}, \,\right\}\mathclose{}$$$. Show that the closure of $$$K$$$ (in the operator norm) is $$$\mathcal{K}\mathopen{}\left( \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathbb{R}\right)\mathclose{}\right)\mathclose{}$$$.

Exercise IV.53

Let $$$S$$$ be the unilateral shift on $$$\mathrm{l}^{0}$$$. That is, $$$S\mathopen{}\left( \mathbf{a}\right)\mathclose{}= S\mathopen{}\left( {a}_{1}, {a}_{2}, {a}_{3}, \dotsc\right)\mathclose{}= \mathopen{}\left(0, {a}_{1}, {a}_{2}, \dotsc\right)\mathclose{}$$$. Show that $$$\mathop{\sigma}\mathopen{}\left( S+ S^{*}\right)\mathclose{}= \mathopen{}\left[{-}2, 2\right]\mathclose{}$$$. (Advice: The inclusion $$$\mathop{\sigma}\mathopen{}\left( S+ S^{*}\right)\mathclose{}\subseteq \mathopen{}\left[{-}2, 2\right]\mathclose{}$$$ is clear. To show $$$\mathopen{}\left[{-}2, 2\right]\mathclose{}\subseteq \mathop{\sigma}\mathopen{}\left( S+ S^{*}\right)\mathclose{}$$$, try solving $$$\mathopen{}\left(S+ S^{*}-λ\right)\mathclose{}\mathopen{}\left( \mathbf{a}\right)\mathclose{}= \mathopen{}\left(1, 0, 0, \dotsc\right)\mathclose{}$$$, say, for $$$\mathbf{a}\in \mathrm{l}^{0}$$$. The algebraic solution of the second-order linear recurrence you encounter is governed by the eigenvalues and eigenvectors of the two-by-two matrix relating $$$\mathopen{}\left({a}_{n+2}, {a}_{n+1}\right)\mathclose{}$$$ to $$$\mathopen{}\left({a}_{n+1}, {a}_{n}\right)\mathclose{}$$$.)

Exercise IV.54

This exercise has to do with the order $$$0$$$ Bessel function of the first kind, that is, the function $$${J}_{0}$$$ given as a power series by $$${J}_{0}\mathopen{}\left( x\right)\mathclose{}= \sum_{n=0}^{\infty}{} \frac{{\mathopen{}\left({-}1\right)\mathclose{}}^{n}}{{\mathopen{}\left({n!}\right)\mathclose{}}^{2}}{\mathopen{}\left(\frac{x}{2}\right)\mathclose{}}^{2n}$$$. It is up to scalar multiple the only solution without singularity at $$$x= 0$$$ of Bessel's equation of order $$$0$$$: $$$xy'' \mathopen{}\left( x\right)\mathclose{}+y' \mathopen{}\left( x\right)\mathclose{}+xy\mathopen{}\left( x\right)\mathclose{}= 0$$$. The main thing here is to show that the sum of the squared reciprocals of the positive zeros of $$${J}_{0}$$$ is $$$\frac{1}{4}$$$. Along the way, an orthonormal basis for $$$\mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$ will emerge that is indispensable for modeling the small oscillations of a flexible chain hung from a hook. (See Chapter 9 of Young's book .)

(a)  Let $$${D}_{0}= \mathopen{}\left\{\, f\in \mathrm{C}^{1}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}\,\middle\vert\, , x\mapsto xf' \mathopen{}\left( x\right)\mathclose{}\in \mathrm{C}^{1}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}, , f\mathopen{}\left( 1\right)\mathclose{}= 0, \,\right\}\mathclose{}$$$. Define $$$L : {D}_{0} \to \mathrm{C}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$ by $$\mathopen{}\left(L\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= {-}xf'' \mathopen{}\left( x\right)\mathclose{}-f' \mathopen{}\left( x\right)\mathclose{}= {-} \frac{\mathrm{d}}{\mathrm{d}x}\mathopen{}\left( xf' \mathopen{}\left( x\right)\mathclose{}\right)\mathclose{} \text{.}$$ Show that the eigenvalues of $$$L$$$ are $$$\mathopen{}\left\{\, \frac{{β}^{2}}{4}\,\middle\vert\, , {J}_{0}\mathopen{}\left( β\right)\mathclose{}= 0, \,\right\}\mathclose{}$$$, with corresponding eigenfunctions $$$x\mapsto {J}_{0}\mathopen{}\left( β\sqrt{x}\right)\mathclose{}$$$. (Advice: $$$L\mathopen{}\left( f\right)\mathclose{}= λf$$$ makes $$$λ{\mathopen{}\left\lVert{}f\right\rVert\mathclose{}}^{2}= \mathopen{}\left\langle{}L\mathopen{}\left( f\right)\mathclose{}, f\right\rangle\mathclose{}$$$. Integrate by parts to show the eigenvalues are positive. For $$$λ\gt 0$$$, show that $$$xf'' \mathopen{}\left( x\right)\mathclose{}+f' \mathopen{}\left( x\right)\mathclose{}+λf\mathopen{}\left( x\right)\mathclose{}= 0$$$ if and only if $$$y= f\mathopen{}\left( 2\sqrt{λx}\right)\mathclose{}$$$ satisfies Bessel's equation of order $$$0$$$.)

(b)  For $$$ξ\in H= \mathrm{L}^{\mathrm{2}}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$, define $$$K\mathopen{}\left( ξ\right)\mathclose{}$$$ on $$$\mathopen{}\left[0, 1\right]\mathclose{}$$$ by $$\mathopen{}\left(K\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= {-} \log\mathopen{}\left( x\right)\mathclose{} \int _{0}^{x}{}ξ\mathopen{}\left( t\right)\mathclose{}\,\mathrm{d}t-\int _{x}^{1}{} \log\mathopen{}\left( t\right)\mathclose{}ξ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t\text{.}$$ Show that $$$K$$$ is a self-adjoint Hilbert-Schmidt operator on $$$H$$$ mapping $$$\mathrm{C}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$ to $$${D}_{0}$$$, that $$$L\mathopen{}\left( K\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}= ξ$$$ for $$$ξ\in \mathrm{C}\mathopen{}\left( \mathopen{}\left[0, 1\right]\mathclose{}\right)\mathclose{}$$$, and that $$$K\mathopen{}\left( L\mathopen{}\left( f\right)\mathclose{}\right)\mathclose{}= f$$$ for $$$f\in {D}_{0}$$$. (Advice: Exhibit $$$K$$$ as an integral operator with kernel $$$k\in \mathrm{L}^{\mathrm{2}}\mathopen{}\left( {\mathopen{}\left[0, 1\right]\mathclose{}}^{2}\right)\mathclose{}$$$ satisfying $$$k\mathopen{}\left( x, t\right)\mathclose{}= k\mathopen{}\left( t, x\right)\mathclose{}$$$. Notice $$$\log\in H$$$ and use L'Hôpital's Rule at $$$x= 0$$$ when necessary.)

(c)  Let $$${β}_{i}$$$, $$$i\in \mathopen{}\left\{\, 1, 2, \dotsc\,\right\}\mathclose{}$$$, denote the positive zeros of $$${J}_{0}$$$. Define $$${φ}_{j}$$$ on $$$\mathopen{}\left[0, 1\right]\mathclose{}$$$ by $${φ}_{j}\mathopen{}\left( x\right)\mathclose{}= \frac{{J}_{0}\mathopen{}\left( {β}_{j}\sqrt{x}\right)\mathclose{}}{{\mathopen{}\left(\int _{0}^{1}{}{J}_{0}{\mathopen{}\left({β}_{j}\sqrt{x}\right)\mathclose{}}^{2}\,\mathrm{d}x\right)\mathclose{}}^{\frac{1}{2}}}\text{.}$$ Show that $$$\mathopen{}\left\{\, {φ}_{1}, {φ}_{2}, \dotsc\,\right\}\mathclose{}$$$ is an orthonormal basis for $$$H$$$ with $$$K\mathopen{}\left( {φ}_{j}\right)\mathclose{}= \frac{4}{{{β}_{j}}^{2}}{φ}_{j}$$$ for every $$$j$$$. (This includes showing that $$$\operatorname{Ker}\mathopen{}\left( K\right)\mathclose{}= \mathopen{}\left\{\, 0\,\right\}\mathclose{}$$$, and accounting for all of the nonzero eigenvalues of $$$K$$$.)

It is experimentally obvious, and not very difficult to show, that $$${β}_{j}\gt j$$$ for every $$$j$$$. It follows that $$$K$$$ is a trace class operator, with trace $$$\sum_{j}{} {{β}_{j}}^{{-}2}$$$. We would like to identify this number in a more vivid way by integrating the kernel you found in part (b) along the diagonal. Unfortunately, though, the kernel has a singularity at $$$\mathopen{}\left(0, 0\right)\mathclose{}$$$, and proof of the relevant theorem strongly uses uniform continuity of the kernel. To save the calculation, we have to approximate $$$K$$$ by operators with kernels that are continuous everywhere on the square.

(d)  For $$$n\geq 2$$$, $$$n\in \mathbb{N}$$$, let $$${Λ}_{n}$$$ be the function on $$$\mathopen{}\left[0, 1\right]\mathclose{}$$$ that is $$$\log\mathopen{}\left( n\right)\mathclose{}$$$ on $$$\mathopen{}\left[0, \frac{1}{n}\right]\mathclose{}$$$ and $$${-}\log\mathopen{}\left( n\right)\mathclose{}$$$ on $$$\mathopen{}\left(\frac{1}{n}, 1\right]\mathclose{}$$$. Define the operator $$${K}_{n}$$$ by $$\mathopen{}\left({K}_{n}, ξ\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= {Λ}_{n}\mathopen{}\left( x\right)\mathclose{}\int _{0}^{x}{}ξ\mathopen{}\left( t\right)\mathclose{}\,\mathrm{d}t+\int _{x}^{1}{} {Λ}_{n}\mathopen{}\left( t\right)\mathclose{}ξ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t\text{.}$$ You can easily write down the kernel for $$${K}_{n}$$$ and see thereby that $$${K}_{n}$$$ is a self-adjoint Hilbert-Schmidt operator on $$$H$$$. Consider also $$${Δ}_{n}$$$ defined by $$\mathopen{}\left({Δ}_{n}\mathopen{}\left( ξ\right)\mathclose{}\right)\mathclose{}\mathopen{}\left( x\right)\mathclose{}= \begin{cases}0, & x\gt \frac{1}{n} ; \\ {-}\log\mathopen{}\left( x\right)\mathclose{}\int _{0}^{x}{}ξ\mathopen{}\left( t\right)\mathclose{}\,\mathrm{d}t-\int _{x}^{\frac{1}{n}}{} \log\mathopen{}\left( t\right)\mathclose{}ξ\mathopen{}\left( t\right)\mathclose{} \,\mathrm{d}t, & x\leq \frac{1}{n} \end{cases}\text{.}$$ Show that $$$K= {K}_{n}+{Δ}_{n}-\log\mathopen{}\left( n\right)\mathclose{}\chi_{ \mathopen{}\left[0, \frac{1}{n}\right]\mathclose{} }\otimes \chi_{ \mathopen{}\left[0, \frac{1}{n}\right]\mathclose{} }$$$.

(e)  Show that $$${Δ}_{n}$$$ is trace class and calculate its trace in terms of the Bessel zeros $$${β}_{j}$$$. (Advice: Notice that $$${Δ}_{n}$$$ is the operator obtained by rescaling $$$K$$$ from $$$\mathopen{}\left[0, 1\right]\mathclose{}$$$ to $$$\mathopen{}\left[0, \frac{1}{n}\right]\mathclose{}$$$.)

(f)  It now follows that $$${K}_{n}$$$ is trace class. Find its trace by integrating its kernel function along the diagonal.

(g)  Show that $$$\sum_{j}{} {{β}_{j}}^{{-}2} = \frac{1}{4}$$$.