Counterexample Index

A counterexample identifies a missing hypothesis; it does not erase the valid theorem that included that hypothesis. The entries below pair a tempting generalization with the mechanism that defeats it. Follow the link for the constructed operator, state, or dynamics and its complete verification.

False claims and the mechanism that defeats them

False claim	Counterexample mechanism	Canonical treatment
Every symmetric operator is self-adjoint	A differential expression has a larger adjoint domain than the chosen symmetric domain	Symmetric operators
Every dense test space is a core	Hilbert-norm density need not approximate the operator action in graph norm	Dense domains
A bounded operation preserves every observable domain	A sharp spatial projection introduces a jump and leaves the momentum domain	Domain pitfalls
Every self-adjoint observable has an eigenvector basis	Position multiplication has continuous spectrum and no normalizable point eigenvectors	Continuous spectrum without eigenvectors
Compact positive operators have finite trace	The diagonal entries $1/n$ tend to zero but have divergent sum	Compact operators
A positive map is completely positive	Matrix transpose fails positivity after adjoining an entangled ancilla	Completely positive maps
Every algebraic state on $B(\mathcal H)$ is a density operator	Singular states need not preserve increasing projection limits	Normal and singular states
Trace-norm closeness guarantees entropy closeness	Small probability can spread over increasingly many levels	Entropy and trace
Separable normal states always have countable product ensembles	In infinite dimension, trace-norm closure can require more general barycentric decompositions	Separable mixed states
Strongly continuous channels have bounded generators	Unbounded Hamiltonian frequencies prevent uniform small-time convergence	Strong continuity
Formal trace cancellation proves conservation	An unbounded jump intensity can drive infinitely many jumps in finite time	Bounded and unbounded generators
Valid channels at every time form a divisible family	Information erased at one time can reappear later in a recurrent reduced model	Markovianity language

Several failures disappear in finite dimension. All linear operator domains are then the full finite-dimensional space once an everywhere-defined matrix model is chosen; every bounded operator is compact and trace class; and the standard finite-dimensional continuity topologies agree. Finite matrices remain useful tests of positivity and algebraic claims, but they cannot reveal all infinite-dimensional obstructions.

The table also separates unrelated uses of “infinite.” A separable Hilbert space has a countable orthonormal basis, but this does not require every convex decomposition to use only countably many product states. Basis expansions and barycentric representations are different constructions.

Worked counterexample: a right inverse need not be a left inverse

Let $S$ be the unilateral shift on $\ell^2(\mathbb N_0)$, $Se_n=e_{n+1}$. Set $A=S^\dagger$ and $B=S$. Both are bounded and

$$AB=S^\dagger S=I,\qquad BA=SS^\dagger=I-|e_0\rangle\langle e_0|\ne I.$$

Thus $B$ is a right inverse of $A$ but not a two-sided inverse. The map $A$ is surjective and has a one-dimensional kernel; $B$ is injective and has a proper closed range.

This is a bounded example, so unbounded domains are not the cause. The failure comes from infinite dimension: injectivity and surjectivity are no longer equivalent for an endomorphism. Consequently, an argument that constructs only one side of an inverse must check the other side before declaring a resolvent point.

The same shift appears in Stinespring Dilation as an isometry that is not unitary. Here its role is to diagnose an algebraic inversion mistake. A reusable example can answer several questions, provided the conclusion being tested is stated precisely.

Repair a claim with the smallest adequate hypothesis

To repair “symmetric implies self-adjoint,” boundedness on the entire Hilbert space is one sufficient extra assumption, but it would exclude most Hamiltonians of interest. A useful repair for an unbounded problem instead proves equality of adjoint domains or essential self-adjointness on a chosen core.

To repair an entropy-continuity statement, finite dimension is one option. For infinite-dimensional models, a suitable energy constraint and spectral growth condition can retain the intended physics. To repair a formal Lindblad argument, a bounded-generator theorem is sufficient, while a genuinely unbounded model requires its own existence and conservativity hypotheses.

The repair should fit the actual model. Replacing an infinite-dimensional problem by finite matrices does not establish the missing limit theorem.

Exercises

Why the inverse example cannot be finite dimensional. Let $A,B$ be square matrices of the same finite size and assume $AB=I$. Prove $BA=I$ without assuming either matrix is invertible in advance.

Solution

$AB=I$ implies that $B$ is injective: $Bx=0$ gives $x=ABx=0$. An injective endomorphism of a finite-dimensional space is surjective, hence has an inverse. Multiplying $AB=I$ by that inverse gives $A=B^{-1}$ and therefore $BA=I$. The injective-to-surjective step is exactly what fails for the unilateral shift.

Diagnose an approximation claim. A sequence of finite-rank density operators converges in trace norm to a density operator. Which quantities are guaranteed to converge: bounded-observable expectations, arbitrary unbounded energy means, or von Neumann entropies?

Solution

Every bounded-observable expectation converges because its error is at most $\|\rho_n-\rho\|_1\|A\|$. Unbounded means and entropies need additional control; neither follows from trace-norm convergence alone. This matters for quantum-information approximations that compress a state before evaluating an energy or entropy objective.

References

• Engel, Klaus-Jochen, and Rainer Nagel. One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics 194, Springer, 2000. doi:10.1007/b97696. Continuity and generator distinctions.
• Hall, Brian C. Quantum Theory for Mathematicians. Graduate Texts in Mathematics 267, Springer, 2013. doi:10.1007/978-1-4614-7116-5. Operator and spectral counterexamples.
• Holevo, Alexander S., Maksim E. Shirokov, and Reinhard F. Werner. “Separability and entanglement-breaking in infinite dimensions.” arXiv:quant-ph/0504204 (2005). Preprint. Limits of countable separable decompositions.