Continuous Linear Operators on Hilbert Spaces

Let $[a, b]$ be a closed, bounded interval of real numbers. Show that the $L^{2}[a, b]$ inner-product is also an inner-product on $C[a, b]$. Is $C[a, b]$, considered as an inner-product space with the $L^{2}[a, b]$ inner-product, a Hilbert space?

Show that the maximum norm on $C[a, b]$ is not induced by an inner-product and neither is the usual norm on $\ell^{1}$.

Let $H_{1}$ and $H_{2}$ be Hilbert spaces. Show that the Cartesian product $H_{1} \times H_{2}$ also is a Hilbert space with an inner-product with respect to which $H_{1} \times \left\{ 0\right\} = \left[\left\{ 0\right\} \times H_{2}\right]^{\perp }$.

Show that if $S$ is a subset of an inner-product space $H$, then $S^{\perp }$ is a closed subspace of $H$.

Let $S$ be a subset of $H$. Show that $S = \left(S^{\perp }\right)^{\perp }$ if and only if $S$ is a closed subspace of $H$.

Show that for any two vectors $u, v \in H$,

Let $X$ be a linear space normed by $\left\| \cdot \right\|$. Use the polarization identity to show that a norm $\left\| \cdot \right\|$ is induced by an inner-product if and only if the parallelogram identity holds.

Let $V$ be a closed subspace of $H$ and $P$ a projection of $H$ onto $V$. Show that $P$ is the orthogonal projection of $H$ onto $V$ if and only if (4) holds.

Let $T$ belong to $\mathcal{L}(H)$. Show that $T$ is an isometry if and only if

Let $V$ be a finite dimensional subspace of $H$ and $\varphi_{1}, \ldots , \varphi_{n}$ a basis for $V$ consisting of unit vectors, each pair of which is orthogonal. Show that the orthogonal projection $P$ of $H$ onto $V$ is given by

For $h$ a vector in $H$, show that the function $u \mapsto \left\langle h, u \right\rangle$ belongs to $H^{*}$.

For any vector $h \in H$, show that there is a bounded linear functional $\psi \in H^{*}$ for which

Let $V$ be a closed subspace of $H$ and $P$ the orthogonal projection of $H$ onto $V$. For any normed linear space $X$ and $T \in \mathcal{L}(V, X)$, show that $T \circ P$ belongs to $\mathcal{L}(H, X)$, and is an extension of $T: V \rightarrow X$ for which $\left\| T \circ P\right\| = \left\| T\right\|$.

Prove the Hyperplane Separation Theorem for $H$, considered as a locally convex topological vector space with respect to the strong topology, by directly using Proposition 1.

Proposition 1: Let $K$ be a non-empty, closed, convex subset of a Hilbert space $H$ and $h_{0}$ be a vector in $H$ that lies outside of $K$. Then there is exactly one vector $h^{*} \in K$ that is closest to $h_{0}$, in the sense that $\left\| h_{0} - h^{*}\right\| = \operatorname {dist}_{K}(h_{0}) = \inf_{h \in K} \left\| h_{0} - h\right\|$.

Use Proposition 1 to prove the Krein-Milman Lemma in a Hilbert space.

Proposition 1: Let $K$ be a non-empty, closed, convex subset of a Hilbert space $H$ and $h_{0}$ be a vector in $H$ that lies outside of $K$. Then there is exactly one vector $h^{*} \in K$ that is closest to $h_{0}$, in the sense that $\left\| h_{0} - h^{*}\right\| = \operatorname {dist}_{K}(h_{0}) = \inf_{h \in K} \left\| h_{0} - h\right\|$.

Show that an orthonormal subset of a separable Hilbert space $H$ must be countable.

Let $\left\{ \varphi_{k}\right\}$ be an orthonormal sequence in a Hilbert space $H$. Show that $\left\{ \varphi_{k}\right\}$ converges weakly to 0 in $H$.

Let $\left\{ \varphi_{k}\right\}$ be an orthonormal basis for the separable Hilbert space $H$. Show that $\left\{ u_{n}\right\} \rightarrow u$ weakly if and only if $\left\{ u_{n}\right\}$ is bounded and, for each $k, \lim_{n \rightarrow \infty } \left\langle u_{n}, \varphi_{k} \right\rangle = \left\langle u, \varphi_{k} \right\rangle$.

Show that any two infinite dimensional separable Hilbert spaces are isometrically isomorphic and that any such isomorphism preserves the inner-product.

Let $V$ a closed, separable subspace of $H$ for which $\left\{ \varphi_{k}\right\}$ is an orthonormal basis. Show that the orthogonal projection of $H$ onto $V, P$, is given by

Let $\left\{ \varphi_{k}\right\}$ be an orthonormal basis for a Hilbert space $H$. Verify that

Also verify that

where, for each natural number $k, a_{k} = \left\langle u, \varphi_{k} \right\rangle$ and $b_{k} = \left\langle v, \varphi_{k} \right\rangle$.

Verify the assertions in the example of the orthonormal basis for $L^{2}[0,2 \pi ]$.

Use Proposition 5 and the Stone-Weierstrass Theorem to show that for each $f \in L^{2}[-\pi , \pi ]$,

where the convergence is in $L^{2}[-\pi , \pi ]$ and each

Proposition 5: An orthonormal sequence $\left\{ \varphi_{k}\right\}$ in a Hilbert space $H$ is complete if and only if it is an orthonormal basis.

Show that neither $\ell^{1}, \ell^{\infty }, L^{1}[a, b]$ nor $L^{\infty }[a, b]$ is Hilbertable.

Let $H$ be an inner-product space. Show that since $H$ is a dense subset of a Banach space $X$ whose norm restricts to the norm induced by the inner-product on $H$, the inner-product on $H$ extends to $X$ and induces the norm on $X$. Therefore, inner-product spaces have Hilbert space completions.

The proof is based on the RieszFréchet Representation Theorem. Let ($X, \mathcal{M}, \mu$) be a finite measure space, and $\nu : \mathcal{M} \rightarrow [0, \infty )$ be a finite measure that is absolutely continuous with respect to $\mu$. Fill in the following outline of the proof that there is a non-negative measurable function $h: X \rightarrow [0, \infty )$ for which $\nu (E) = \int_{E} h d \mu$ for all $E \in \mathcal{M}$.

Verify (14).

Let $T, S \in \mathcal{L}(H)$ be symmetric. Show that $T = S$ if and only if $Q_{T} = Q_{S}$.

Show the bounded, symmetric operators are a closed subspace of $\mathcal{L}(H)$. Also show that if $T$ and $S$ are symmetric, then so is the composition $S \circ T$ if and only if $T$ commutes with $S$ with respect to composition, that is, $S \circ T = T \circ S$.

Let $H$ be a Hilbert space and the linear operator $T: H \rightarrow H$ have the property that $\left\langle T(u), v \right\rangle = \left\langle u, T(v) \right\rangle$ for all $u, v \in H$. Show that $T$ belongs to $\mathcal{L}(H)$, that is, $T$ is bounded.

Exhibit an operator $T \in \mathcal{L}\left(\mathbb {R}^{2}\right)$ for which $\left\| T\right\| > \sup_{\left\| u\right\| = 1}\left| \left\langle T(u), u \right\rangle \right|$.

Let $S, T \in \mathcal{L}(H)$ be symmetric. Assume that $S \geq T$ and $T \geq S$. Prove that $T = S$.

Let $V$ be a closed, nontrivial subspace of a Hilbert space $H$ and $P$ the orthogonal projection of $H$ onto $V$. Show that $P = P^{*}, P \geq 0$, and $\left\| P\right\| = 1$.

Let $P \in \mathcal{L}(H)$ be a projection. Show that $P$ is the orthogonal projection of $H$ onto $P(H)$ if and only if $P = P^{*}$.

Let $\left\{ \varphi_{k}\right\}$ be an orthonormal basis for a Hilbert space $H$ and for each natural number $n$, define $P_{n}$ to be the orthogonal projection of $H$ onto the linear span of $\left\{ \varphi_{1}, \ldots , \varphi_{n}\right\}$. Show that $P_{n}$ is symmetric and

Show that $\left\{ P_{n}\right\}$ converges pointwise on $H$ to Id but does not converge uniformly on the unit ball.

Show that if $T \in \mathcal{L}(H)$ is invertible, so is $T^{*} \circ T$ and therefore so is $T^{*}$.

Let $T \in \mathcal{L}(H)$ be symmetric and non-negative. Show that for all $u, v \in H$,

Use the preceding problem to show that if $S, T \in \mathcal{L}(H)$ are symmetric and $S \geq T$, then for each $u \in H$, $\left\| S(u)-T(u)\right\|^{4} = \left\langle (S-T)(u), (S-T)(u) \right\rangle^{2} \leq \left| \left\langle (S-T)(u), u \right\rangle \right|\left| \left\langle (S-T)^{2}(u), (S-T)(u) \right\rangle \right|$ and thereby conclude that

A sequence $\left\{ T_{n}\right\}$ of symmetric operators in $\mathcal{L}(H)$ is said to be monotone increasing provided that $T_{n+1} \geq T_{n}$ for each $n$, and said to be bounded above provided that there is a symmetric operator $S$ in $\mathcal{L}(H)$ such that $T_{n} \leq S$ for all $n$.

Let $S \in \mathcal{L}(H)$ be a symmetric operator for which $0 \leq S \leq \operatorname {Id}$. Define a sequence $\left\{ T_{n}\right\}$ in $\mathcal{L}(H)$ by letting $T_{1} = 1 / 2(\operatorname {Id}-S)$ and if $n$ is a natural number for which $T_{n} \in \mathcal{L}(H)$ has been defined, defining $T_{n+1} = 1 / 2\left(\mathrm{Id}-S+T_{n}^{2}\right)$.

Let $T \in \mathcal{L}(H)$ be a non-negative, symmetric operator. A non-negative, symmetric operator $A \in \mathcal{L}(H)$ is called a square root of $T$ provided that $A^{2} = T$. Use the inductive construction in the preceding problem to show that $T$ has a square root $A$ which commutes with each operator in $\mathcal{L}(H)$ that commutes with $T$. Show that the square root is unique: it is denoted by $\sqrt{T}$. Finally, show that $T$ is invertible if and only if $\sqrt{T}$ is invertible.

An invertible operator $T \in \mathcal{L}(H)$ is said to be orthogonal provided that $T^{-1} = T^{*}$. Show that an invertible operator is orthogonal if and only if it is an isometry.

Let $T \in \mathcal{L}(H)$ be invertible. Show that there is an orthogonal invertible operator $A \in \mathcal{L}(H)$ and a non-negative, symmetric invertible operator $B \in \mathcal{L}(H)$ such that $T = B \circ A$.

Show that if $H$ is infinite dimensional and $T \in \mathcal{L}(H)$ is invertible, then $T$ is not compact.

Prove Proposition 17.

Proposition 17: For $T \in \mathcal{L}(H)$, the following three properties are equivalent: (i) $T$ is compact; (ii) $T(B)$ is totally bounded, where $B$ is the closed unit ball in $H$; (iii) if $\left\{ h_{n}\right\}$ is a bounded sequence in $H$, then $\left\{ T(h_{n})\right\}$ has a strongly convergent subsequence.

Let $\mathcal{K}(H)$ denote the set of compact operators in $\mathcal{L}(H)$. Show that $\mathcal{K}(H)$ is a linear subspace of $\mathcal{L}(H)$. Moreover, show that for $K \in \mathcal{K}(H)$ and $T \in \mathcal{L}(H)$, both $K \circ T$ and $T \circ K$ belong to $\mathcal{K}(H)$.

Show that a linear operator $T: H \rightarrow H$ is continuous if and only if it maps weakly convergent sequences to weakly convergent sequences.

Show that $K \in \mathcal{L}(H)$ is compact if and only if whenever if $\left\{ u_{n}\right\} \rightarrow u$ weakly and $\left\{ v_{n}\right\} \rightarrow v$ weakly, then $\left\langle K\left(u_{n}\right), v_{n} \right\rangle \rightarrow \left\langle K(u), v \right\rangle$.

Let $\left\{ P_{n}\right\}$ be a sequence of orthogonal projections in $\mathcal{L}(H)$ with the property that for natural numbers $n$ and $m, P_{n}(H)$ and $P_{m}(H)$ are orthogonal finite dimensional subspaces of $H$. Let $\left\{ \lambda_{n}\right\}$ be a bounded sequence of real numbers. Show that

is a properly defined symmetric operator in $\mathcal{L}(H)$ that is compact if and only if $\left\{ \lambda_{n}\right\}$ converges to 0.

For $X$ a Banach space, define an operator $T \in \mathcal{L}(X)$ be compact provided that $T(B)$ has compact closure. Show that Proposition 17 holds for a general Banach space and Proposition 18 holds for a reflexive Banach space.

Proposition 17: For $T \in \mathcal{L}(H)$, the following three properties are equivalent: (i) $T$ is compact; (ii) $T(B)$ is totally bounded, where $B$ is the closed unit ball in $H$; (iii) if $\left\{ h_{n}\right\}$ is a bounded sequence in $H$, then $\left\{ T(h_{n})\right\}$ has a strongly convergent subsequence. Proposition 18: An operator $T$ in $\mathcal{L}(H)$ is compact if and only if it maps weakly convergent sequences to strongly convergent sequences, that is, if $\left\{ h_{n}\right\} \rightarrow h$ weakly, then $\left\{ T(h_{n})\right\} \rightarrow T(h)$ strongly.

Let $T \in \mathcal{L}(H)$ be compact and symmetric. Define

Show that if $\alpha < 0$, then $\alpha$ is an eigenvalue of $T$ and if $\beta > 0$, then $\beta$ is an eigenvalue of $T$. Exhibit an example where $\alpha = 0$ and yet $\alpha$ is not an eigenvalue of $T$, that is, $T$ is one-to-one.

Let $K \in \mathcal{L}(H)$ be compact and symmetric. Suppose

Let $\left\{ h_{n}\right\}$ be a sequence of unit vectors for which $\lim_{n \rightarrow \infty } \left\langle K\left(h_{n}\right), h_{n} \right\rangle = \beta$. Show that a subsequence of $\left\{ h_{n}\right\}$ converges strongly to an eigenvector of $T$ with corresponding eigenvalue $\beta$.

Let $K \in \mathcal{L}(H)$ be compact. Show that $T = K^{*} K$ is compact and symmetric. Then use the Hilbert-Schmidt Theorem to show that there is an orthonormal sequence $\left\{ \varphi_{k}\right\}$ of $H$ such that $T\left(\varphi_{k}\right) = \lambda_{k} \varphi_{k}$ for all $k$ and $T(h) = 0$ if $h$ is orthogonal to $\left\{ \varphi_{k}\right\}_{k = 0}^{\infty }$. Conclude that if $h$ is orthogonal to $\left\{ \varphi_{k}\right\}_{k = 0}^{\infty }$, then

Define $H_{0}$ to be the closed linear span of $\left\{ K^{m}\left(\varphi_{k}\right) \mid m \geq 0, k \geq 1\right\}$. Show that $H_{0}$ is closed and separable, $K\left(H_{0}\right) \subseteq H_{0}$ and $K = 0$ on $H_{0}^{\perp }$.

Let $\mathcal{K}(H)$ denote the set of compact operators in $\mathcal{L}(H)$. Show that $\mathcal{K}(H)$ is a closed subspace of $\mathcal{L}(H)$ that has the set of operators of finite rank as a dense subspace. Is $\mathcal{K}(H)$ an open subset of $\mathcal{L}(H)$ ?

Show that the composition in either order of a Fredholm operator of index 0 with an invertible operator is also Fredholm of index 0.

Show that the composition of two Fredholm operators of index 0 is also Fredholm of index 0.

Show that an operator $T \in \mathcal{L}(H)$ is Fredholm of index 0 if and only if it is the perturbation of an invertible operator by an operator of finite rank.

Argue as follows to show that the collection of invertible operators in $\mathcal{L}(H)$ is an open subset of $\mathcal{L}(H)$.

Show that the set of operators in $\mathcal{L}(H)$ that are Fredholm of index 0 is an open subset of $\mathcal{L}(H)$.

By following the orthogonal approximation sequence method used in the proof of Proposition 22, provide another proof of Lemma 14 in case $H$ is separable.

For $T \in \mathcal{L}(H)$, suppose that $\left\langle T(h), h \right\rangle \geq \left\| h\right\|^{2}$ for all $h \in H$. Assume that $K \in \mathcal{L}(H)$ is compact and $T+K$ is one-to-one. Show that $T+K$ is onto.

Let $K \in \mathcal{L}(H)$ be compact and $\mu \in \mathbb {R}$ have $\left|\mu \right| > \left\| K\right\|$. Show that $\mu -K$ is invertible.

Let $S \in \mathcal{L}(H)$ have $\left\| S\right\| < 1, K \in \mathcal{L}(H)$ be compact and $(\operatorname {Id}+S+K)(H) = H$. Show that Id $+S+K$ is one-to-one.

Let $\mathcal{G} L(H)$ denote the set of invertible operators in $\mathcal{L}(H)$.

Let $T \in \mathcal{L}(H)$ be Fredholm of index zero, and $K \in \mathcal{L}(H)$ be compact. Show that $T+K$ is Fredholm of index zero.

Let $X_{0}$ be a finite codimensional subspace of a linear space $X$. Show that all finite dimensional linear complements of $X_{0}$ in $X$ have the same dimension.