Bounded Operators on a Hilbert Space

The completion of an inner product space is a Hilbert space. Make this statement more precise, and prove it. (See the proof of Theorem 12.40 for an application).

Suppose $N$ is a positive integer, $\alpha \in \mathcal{C}, \alpha^{N} = 1$, and $\alpha^{2} \neq 1$. Prove that every Hilbert space inner product satisfies the identities

and

Generalize this: Which functions $f$ and measures $\mu$ on a set $\Omega$ give rise to the identity

Let $H^{*}$ be the dual space of $H$; define $\psi : H^{*} \rightarrow H$ by

(See Theorem 12.5). Prove that $H^{*}$ is a Hilbert space, relative to the inner product

If $\phi : H^{* *} \rightarrow H^{*}$ satisfies $z^{* *}\left(y^{*}\right) = \left[y^{*}, \phi z^{* *}\right]$ for all $y^{*} \in H^{*}$ and $z^{* *} \in H^{* *}$, prove that $\psi \phi$ is an isomorphism of $H^{* *}$ onto $H$ whose existence implies that $H$ is reflexive.

Suppose $\left\{ u_{n}\right\}$ is a sequence of unit vectors in $H$ (that is $\left\| u_{n}\right\| = 1$), and assume that

If $\left\{ \alpha_{i}\right\}$ is any sequence of scalars, prove that

and deduce that the following three properties of $\left\{ \alpha_{i}\right\}$ are equivalent to each other:

Suppose $E$ is a resolution of the identity, as in Section 12.17, and prove that

for all $x \in H, y \in H$, and $\omega \in \mathfrak {M}$.

Suppose $U \in \mathscr {B}(H)$ is unitary, and $\varepsilon > 0$. Prove that scalars $\alpha_{0}, \ldots , \alpha_{n}$ can be chosen so that

if $\sigma (U)$ is a proper subset of the unit circle, but that this norm is never less than 1 if $\sigma (U)$ covers the whole circle.

Prove Theorem 12.35 with $P U$ in place of $U P$.

Suppose $T = U P$ is the polar decomposition of an invertible $T \in \mathscr {B}(H)$. Prove that $T$ is normal if and only if $U P = P U$.

Prove that every normal invertible $T \in \mathscr {B}(H)$ is the exponential of some normal $S \in \mathscr {B}(H)$.

Suppose $N \in \mathscr {B}(H)$ is normal, and $T \in \mathscr {B}(H)$ is invertible. Prove that $T N T^{-1}$ is normal if and only if $N$ commutes with $T^{*} T$.

If $T \in \mathscr {B}(H)$ is normal, show that $T^{*} = U T$, for some unitary $U$. When is $U$ unique?

Assume $T \in \mathscr {B}(H)$ and $T^{*} T$ is a compact operator. Show that $T$ is then compact.

Find a noncompact $T \in \mathscr {B}(H)$ such that $T^{2} = 0$. Can such an operator be normal?

Suppose $T \in \mathscr {B}(H)$ is normal, and $\sigma (T)$ is a finite set. Deduce as much information about $T$ from this as you can.

Show, under the hypotheses of $(d)$ of Theorem 12.29, that the equation $T y = x$ has a solution $y \in H$ if and only if

(If $\lambda_{i} = 0$ for one $i$, then $x_{i}$ must be 0, for this $i$).

The spectrum $\sigma (T)$ of $T \in \mathscr {B}(H)$ can be divided into three disjoint pieces:

The point spectrum $\sigma_{p}(T)$ consists of all $\lambda \in \mathbb {C}$ for which $T-\lambda I$ is not one-to-one.

The continuous spectrum $\sigma_{c}(T)$ consists of all $\lambda \in \mathbb {C}$ such that $T-\lambda I$ is a one-to-one mapping of $H$ onto a dense proper subspace of $H$.

The residual spectrum $\sigma_{r}(T)$ consists of all other $\lambda \in \sigma (T)$.

Let $S_{R}$ and $S_{L}$ be as above. Prove that neither $S_{R}$ nor $S_{L}$ has polar decompositions $U P$, with $U$ unitary and $P \geq 0$.

Let $\mu$ be a positive measure on a measure space $\Omega$, let $H = L^{2}(\mu )$, with the usual inner product

For $\phi \in L^{\infty }(\mu )$, define the multiplication operator $M_{\phi }$ by $M_{\phi }(f) = \phi f$. Then $M_{\phi } \in \mathscr {B B}(H)$.

Under what conditions on $\phi$ does $M_{\phi }$ have eigenvalues? Give an example in which $\sigma \left(M_{\phi }\right) = \sigma_{c}\left(M_{\phi }\right)$. Show that every $M_{\phi }$ is normal. What is the relation between $\sigma \left(M_{\phi }\right)$ and the essential range of $\phi$? Show that $\phi \rightarrow M_{\phi }$ is an isometric *-isomorphism of $L^{\infty }(\mu )$ onto a closed subalgebra $A$ of $\mathscr {B}(H)$. (Certain pathological measures $\mu$ have to be excluded in order to make this last statement correct). Is $A$ a maximal commutative sub-algebra of $\mathscr {B}(H)$? Hint: If $T \in \mathscr {B}(H)$ and $T M_{\phi } = M_{\phi } T$ for all $\phi \in L^{\infty }(\mu )$, and if $\mu (\Omega ) < \infty$, show that $T$ is a multiplication by $T(1)$ and hence that $T \in A$.

Suppose $T \in \mathscr {B}(H)$ is normal, $A$ is the closed subalgebra of $\mathscr {B}(H)$ generated by $I$, $T$, and $T^{*}$, and $T$ can be approximated, in the norm topology of $\mathscr {B}(H)$, by finite linear combinations of projections that belong to $A$.

Under what (necessary and sufficient) conditions on $\sigma (T)$ does this happen?

Does every normal $T \in \mathscr {B}(H)$ have a square root in $\mathscr {B}(H)$? What can you say about the cardinality of the set of all square roots of $T$? Can it happen that two square roots of the same $T$ do not commute? Can this happen when $T = I$?

Show that the Fourier transform $f \rightarrow \hat{f}$ is a unitary operator on $L^{2}\left(\mathbb {R}^{n}\right)$. What is its spectrum? Suggestion: When $n = 1$, compute the Fourier transforms of

Show that any two infinite-dimensional separable Hilbert spaces are isometrically isomorphic (via countable orthonormal bases; see [23]). Show that the space $H$ in Theorem 12.38 is separable. Show that the answer to the question that precedes Theorem 12.38 is therefore negative for every infinite-dimensional $H$, separable or not.

Suppose $T \in \mathscr {B}(H)$ is normal, $f$ is a bounded Borel function on $\sigma (T)$, and $S = f(T)$. If $E_{T}$ and $E_{S}$ are the spectral decompositions of $T$ and $S$, respectively, prove that

for every Borel set $\omega \subset \sigma (S)$.

If $S \in \mathscr {B}(H)$ and $T \in \mathscr {B}(H)$, the notation $S \geq T$ means that $S-T \geq 0$, that is, that

for all $x \in H$. Prove the equivalence of the following four properties of a pair of self-adjoint projections $P$ and $Q$ :

Suppose * is an involution in a complex algebra $A, q$ is an invertible element of $A$ such that $q^{*} = q$ and $x^{*}$ is defined by

for every $x \in A$. Show that ${ }^{*}$ is an involution in $A$.

Let $A$ be the algebra of all complex 4 -by-4 matrices. If $M = \left(m_{i j}\right) \in A$, let $M^{*}$ be the conjugate transpose of $M: m_{i j}^{*} = \overline{m_{j i}}$. Put

As in Exercise 27, define

Let $X$ be the vector space of all trigonometric polynomials on the real line: these are functions of the form

where $s_{k} \in R$ and $c_{k} \in \mathscr {C}$, for $1 \leq k \leq n$. Show that

exists and defines an inner product on $X$, that

and that the completion of $X$ is a nonseparable Hilbert space $H$. Show that $H$ contains all uniform limits of trigonometric polynomials; these are the so-called " almost-periodic" functions on $R$.

Let $H_{w}$ be an infinite-dimensional Hilbert space, with its weak topology. Prove that the inner product is a separately continuous function on $H_{w} \times H_{w}$ which is not jointly continuous.

Assume $T_{n} \in \mathscr {B}(H)$ for $n = 1,2,3, \ldots$, and

for every $x \in H$. Does it follow that

for every $x \in H$?

Let $X$ be a uniformly convex Banach space. This means, by definition, that the assumptions

imply that $\left\| x_{n}-y_{n}\right\| \rightarrow 0$. For example, every Hilbert space is uniformly convex.

Prove the assertion about the case $\operatorname {dim} H < \infty$ made in the remark that follows Theorem 12.35.

Find an operator $T \in \mathscr {B}(H)$, with $\sigma (T) = \left\{ 1\right\}$, which is neither unitary nor selfadjoint.

If $S$ is self-adjoint and $U = \exp (i S)$, show that $U$ is unitary. Deduce from this, and from the fact that $\sigma (U)$ lies on the unit circle, that $\sigma (S)$ lies on the real axis.

Show that $\mathscr {R}\left(T^{*}\right) = \mathscr {R}(T)$ if $T \in \mathscr {B}(H)$ is normal. Hint: Using Theorem 12.35, $T = T^{*} U^{2}$.

Define $T$ on $H = L^{2}(0,1)$ by $(T f)(x) = x f(x)$. Show that $T$ is self-adjoint and that $\mathscr {R}(T)$ is a dense proper subspace of $H$.

Find a nonnormal $T \in \mathscr {B}(H)$ such that

(This shows that Theorem 12.25 has no converse).

Show that $T$ and $T^{*}$ can have the same null space without being normal.

Let $D$ be a circular annulus in $\mathbb {C}$ with center at 0. Define $H$ and $M \in \mathscr {B}(H)$ as in Theorem 12.38. Prove that $M$ has no square root in $\mathscr {B}(H)$ by completing the following outline: Assume $Q \in \mathscr {B}(H), Q^{2} = M$. Put $u(z) = 1, v(z) = z, h = Q u$. Since $Q M = M Q$, induction shows that $Q v^{n} = h v^{n}$ for all integers $n$. It follows from the Laurent series expansion that $Q f = h f$ for every $f \in H$. This leads to $h^{2} = v$, i.e., $h^{2}(z) = z$ for all $z \in D$, an impossibility.

Find the adjoint $M^{*}$ of $M$. (Use Laurent series).