Bounded Operators: Norms, Extension, and Adjoints

A bounded operator changes vector norms by at most a fixed factor. This property makes continuity, adjoints, products, and extension from dense subspaces behave much like matrix algebra. Boundedness does not imply finite rank, compactness, or a discrete spectrum; those are separate restrictions.

Helpful background. Operators in Quantum Mechanics distinguishes operators used as transformations, observables, and generators.

Operator norm and continuity

For a linear map $B:\mathcal H\to\mathcal K$ between Hilbert spaces, define

$$\|B\|=\sup_{\|\psi\|=1}\|B\psi\|.$$

The operator is bounded when this supremum is finite. Equivalently, $\|B\psi\|\leq C\|\psi\|$ for every $\psi$ and some finite $C$. The smallest admissible constant is the operator norm.

Boundedness is equivalent to norm continuity. One direction follows from

$$\|B\psi-B\phi\|\leq\|B\|\|\psi-\phi\|.$$

Conversely, continuity at zero gives a $\delta>0$ such that $\|\psi\|<\delta$ implies $\|B\psi\|<1$. Rescaling an arbitrary nonzero vector by a factor smaller than $\delta/\|\psi\|$ produces a uniform linear bound. Linearity makes a local continuity statement control the whole space.

The collection $\mathcal B(\mathcal H)$ is a Banach algebra under composition:

$$\|BC\|\leq\|B\|\|C\|, \qquad \|B+C\|\leq\|B\|+\|C\|.$$

Completeness means that an operator-norm Cauchy sequence converges to a bounded operator. This is what legitimizes norm-convergent series such as the exponential of a bounded Hamiltonian.

Extending an action from a dense domain

Suppose $B_0$ is initially defined on a dense linear subspace $D$ and obeys a uniform estimate $\|B_0\psi\|\leq C\|\psi\|$ there. For any $\psi\in\mathcal H$, choose $\psi_n\in D$ with $\psi_n\to\psi$ and define

$$B\psi=\lim_{n\to\infty}B_0\psi_n.$$

The estimate makes the image sequence Cauchy. If another approximating sequence is used, the difference of its images tends to zero, so the result is independent of the choice. This constructs a unique bounded extension to all of $\mathcal H$.

The uniform estimate is indispensable. Mere density of $D$ does not permit extending an unbounded action by continuity. Also, the initial restriction $B_0$ remains an operator with domain $D$; it is not literally identical to its everywhere-defined extension. Statements such as “a bounded symmetric operator is self-adjoint” normally include the full-domain convention.

For example, a projection first specified on finite linear combinations of an orthonormal basis extends continuously to the whole Hilbert space. A diagonal action $e_n\mapsto n e_n$ has no finite bound and cannot be extended by this argument.

Adjoints and elementary quantum examples

The Riesz representation theorem gives a unique bounded adjoint $B^*:\mathcal K\to\mathcal H$ satisfying

$$\langle\phi,B\psi\rangle=\langle B^*\phi,\psi\rangle.$$

Here the inner product is linear in its second argument. The norm identities

$$\|B^*\|=\|B\|, \qquad \|B^*B\|=\|B\|^2$$

follow from the dual characterization of the vector norm and from $\|B\psi\|^2=\langle\psi,B^*B\psi\rangle$. These identities explain why positivity and operator norm fit naturally together.

For vectors $u,v$, the rank-one operator $B\psi=u\langle v,\psi\rangle$ has norm $\|u\|\|v\|$. Cauchy–Schwarz gives the upper bound, and a unit vector proportional to $v$ attains it when $v\neq0$. A normalized rank-one projection therefore has norm one.

On $L^2(X,\mu)$ for a sigma-finite measure space, multiplication by an essentially bounded measurable function $f$ obeys

$$\|M_f\|=\|f\|_\infty.$$

The upper bound follows by integrating $|f\psi|^2$. To obtain the reverse bound, choose normalized vectors supported where $|f|$ lies arbitrarily close to its essential supremum. On the usual sigma-finite measure spaces this support can be chosen to have finite positive measure. The essential supremum, rather than a pointwise maximum, is required because null sets do not change an $L^2$ operator.

Strong convergence is weaker than norm convergence

Let $P_N$ project onto the first $N$ vectors of an infinite orthonormal basis. Then

$$\|P_N\psi-\psi\|^2=\sum_{n>N}|\langle e_n,\psi\rangle|^2\longrightarrow0$$

for each fixed $\psi$. Thus $P_N\to I$ strongly. Nevertheless $\|I-P_N\|=1$ for every $N$, because a unit vector in the omitted subspace is changed by norm one.

A finite-dimensional truncation can therefore converge for every fixed state without controlling the worst possible state uniformly. In numerical quantum mechanics, a convergence claim should identify which topology and which set of states the error estimate controls. This example also shows that the identity is not compact on an infinite-dimensional Hilbert space.

Exercises

Let $B$ be bounded with $\|B\|<1$. Show that $I-B$ has a bounded inverse and obtain a norm estimate.

Solution

The series $\sum_{n=0}^\infty B^n$ converges in operator norm because $\sum\|B\|^n$ converges. Multiplying a partial sum by $I-B$ gives $I-B^{N+1}$; passing to the norm limit gives the inverse on either side. Its norm is at most $(1-\|B\|)^{-1}$. This Neumann-series argument underlies the openness of the resolvent set.

Why can a bounded multiplication observable on $L^2([0,1])$ have no normalizable eigenvectors?

Solution

For $(B\psi)(x)=x\psi(x)$, an eigenvector with eigenvalue $\lambda$ must vanish except on the singleton $\{\lambda\}$. That set has Lebesgue measure zero, so the only such $L^2$ vector is zero. Boundedness controls continuity of the action, not the existence of an eigenbasis. The spectrum is the interval $[0,1]$.

References

• R. B. Melrose, Functional Analysis, Massachusetts Institute of Technology lecture notes, 2020, course reading page — bounded operators, adjoints, and compactness.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, doi:10.1007/978-94-007-4753-1 — bounded-operator background and contrasts with unbounded operators.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., American Mathematical Society, 2014, author’s book page — Hilbert-space preliminaries and multiplication operators.