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A solution to the inverse generator problem and related questions

This paper resolves the inverse generator problem on Hilbert spaces in the negative by constructing bounded operators that generate stable $C_0$ -semigroups while their inverses fail to generate semigroups or produce unbounded inverse semigroups, thereby demonstrating the instability of the Crank–Nicolson scheme and the limitations of the Kreiss resolvent condition through finite-dimensional Schauder multiplier constructions.

Original authors: Emiel Lorist, Martin Meyries, Mark Veraar

Published 2026-08-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Emiel Lorist, Martin Meyries, Mark Veraar

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of mathematics, there is a special class of tools used to describe how things change over time, from the cooling of a cup of coffee to the movement of planets. These tools are called semigroups, and they rely on a central engine known as a generator. Think of the generator as the master switch that dictates the speed and direction of the change. For decades, mathematicians have been fascinated by a specific puzzle involving these generators: if you have a machine that runs smoothly and predictably, does its reverse operation also run smoothly? In technical terms, if a generator creates a stable, bounded flow of change, does its mathematical inverse create a similar stable flow? This question, known as the inverse generator problem, has been a subject of intense study because the answer matters for how we simulate complex systems on computers. If the reverse operation behaves badly, it can cause computer models to spiral out of control, producing garbage results instead of accurate predictions.

For a long time, the answer seemed to be yes, at least in the most well-behaved mathematical spaces. However, a new study by researchers Emiel Lorist, Martin Meyries, and Mark Veraar has definitively shown that this intuition is wrong. They have constructed a specific, explicit example of a mathematical machine that runs perfectly well on its own but falls apart when you try to run it in reverse. Their work proves that on a Hilbert space—a type of mathematical environment used to model many physical systems—it is possible to have a generator that produces a stable, bounded flow of time, yet its inverse does not generate a stable flow at all. In fact, the reverse process grows so wildly that it becomes unbounded, meaning the numbers involved can become infinitely large. This discovery settles a major open question in the field, proving that the relationship between a system and its reverse is far more fragile than previously thought.

The researchers did not just find a theoretical possibility; they built a concrete counterexample using a clever construction involving finite-dimensional matrices. They created a sequence of increasingly complex matrices that act as the building blocks for their final machine. These matrices were designed with very specific properties: their internal parts are arranged so that when the machine runs forward, the energy stays under control and eventually fades away. However, the arrangement is so delicate that when the machine is reversed, the energy does not just stay stable; it explodes. The growth of this reverse process is not merely fast; it follows a very specific, slow-burning pattern that the authors describe as growing at least double logarithmically. To put this in perspective, while a normal explosion might grow like a square or a cube, this growth is incredibly slow at first but eventually becomes unbounded, defying the expectation that a stable system should have a stable reverse.

This finding has immediate and serious consequences for how we solve equations on computers. One of the most popular methods for simulating time-dependent systems is called the Crank–Nicolson scheme. It is a standard tool used in engineering and physics because it is usually very stable and accurate. The new study shows that for the specific type of system they constructed, this popular method fails completely. No matter how small the time steps are made, or how long the simulation runs, the computer model will eventually diverge. The error does not just stay small; it grows without limit. The researchers demonstrated that the instability is not a minor glitch but a fundamental feature of the system, proving that the Crank–Nicolson scheme is not universally stable, even for systems that are exponentially stable in their forward direction.

The study also sheds light on the behavior of a mathematical tool called the Cayley transform, which is closely related to the Crank–Nicolson scheme. This tool is often used to analyze whether a system will remain stable over time. The researchers found that their counterexample satisfies the basic conditions for stability that mathematicians have relied on for years, yet it still fails to be stable in the stronger sense required for long-term calculations. This means that the standard checks used to guarantee stability are not enough to prevent the system from blowing up. The work reveals a hidden layer of complexity in these systems, showing that they can pass all the usual tests for safety while still harboring a potential for catastrophic failure when reversed or simulated.

The construction of this counterexample relied on a deep understanding of how different mathematical bases interact. The researchers used a specific set of vectors that are almost, but not quite, orthogonal to each other. This slight imperfection, which they controlled with a specific parameter, allowed them to tune the system so that the forward motion was perfectly damped while the reverse motion was amplified. The key to their success was using a sequence of numbers that grow at a double-exponential rate. This rapid growth is what allows the system to maintain stability in one direction while becoming unstable in the other. Without this specific, carefully chosen growth rate, the delicate balance required for the counterexample would not exist.

In the end, this paper provides a definitive negative answer to a question that had remained open for years. It shows that the inverse generator problem does not have a positive solution on Hilbert spaces, and that the stability of numerical schemes cannot be taken for granted. The researchers have provided a clear, explicit example that serves as a warning to mathematicians and engineers: just because a system looks stable and passes standard tests, it does not mean its reverse is safe, nor does it mean that standard computer simulations will always work. The work is a rigorous proof, not a suggestion or a simulation, and it stands as a permanent correction to the understanding of these fundamental mathematical relationships.

Technical Summary: A Solution to the Inverse Generator Problem and Related Questions

1. Problem Statement

The paper addresses three closely related open problems concerning the generator $A$ of a $C_0$ -semigroup on a Hilbert space $H$ :

The Inverse Generator Problem: Posed by deLaubenfels, this asks whether the inverse operator $A^{-1}$ (defined on the dense range of $A$ ) generates a bounded $C_0$ -semigroup, given that $A$ generates a bounded $C_0$ -semigroup with dense range. While positive results exist for sectorial operators and contractive semigroups, the general case on Hilbert spaces remained open. Previous counterexamples on Hilbert spaces either involved unbounded original semigroups or only established logarithmic growth bounds for the inverse.
Power Boundedness of Cayley Transforms: For a bounded $C_0$ -semigroup generator $A$ , the Cayley transform is defined as $C_h(A) = (I + \frac{h}{2}A)(I - \frac{h}{2}A)^{-1}$ . It is known that these transforms satisfy the ordinary Kreiss resolvent condition. The question is whether they are necessarily strongly Kreiss bounded or power bounded (i.e., $\sup_m \|C_h(A)^m\| < \infty$ ).
Stability of the Crank–Nicolson Scheme: This scheme approximates the solution to $u' = Au$ via $u_{k+1} = C_h(A)u_k$ . Stability requires uniform boundedness of the powers of the Cayley transform. While stable for contractive semigroups or analytic semigroups, its stability for general bounded generators on Hilbert spaces was an open question.

2. Methodology

The authors construct explicit counterexamples by reducing the infinite-dimensional problems to a finite-dimensional construction. The core of the methodology involves:

Finite-Dimensional Counterexamples: Theorem 1.1 establishes the existence of a sequence of invertible, diagonalizable matrices $(A_n)_{n \ge 1}$ in $M_{2n}(\mathbb{C})$ with specific spectral and norm properties.
Explicit Bases with Controlled Unconditionality: The construction relies on explicit bases $(z_j)_{j=1}^{2n}$ of $\mathbb{C}^{2n}$ derived from the work of Ansorena. These bases possess uniformly bounded partial-sum projections but have unconditionality constants growing as $n^\alpha$ for a chosen $\alpha \in (0, 1)$ .
Schauder Multipliers: The matrices $A_n$ are defined as Schauder multipliers with respect to these bases. Their eigenvalues $(\lambda_k)$ are chosen from a sequence with doubly exponential decay in modulus, specifically $\lambda_k = (-1 + i y_k)^{-1}$ , where $y_k$ is a recursively defined integer sequence ( $y_{k+1} = y_k^2 + 1$ ).
Infinite-Dimensional Extension: The infinite-dimensional operators are constructed as direct sums (or direct integrals) of these finite-dimensional blocks. This allows the authors to transfer the growth properties of the finite-dimensional blocks to the infinite-dimensional setting while maintaining the generator properties.

3. Key Contributions and Results

The Inverse Generator Problem

The paper provides a negative solution to the inverse generator problem on Hilbert spaces.

Strongly Stable Case: There exists a bounded operator $A$ with dense range generating a strongly stable $C_0$ -semigroup such that $A^{-1}$ does not generate any $C_0$ -semigroup.
Exponentially Stable Case: There exists an exponentially stable generator $A$ (with $0 \in \rho(A)$ ) such that the semigroup generated by $A^{-1}$ is unbounded. Specifically, the growth is at least doubly logarithmic:
$\|e^{tA^{-1}}\| \gtrsim (\log \log(t + e^e))^\alpha$
This contrasts with the known logarithmic upper bound $\|e^{tA^{-1}}\| \lesssim \log(e+t)$ for exponentially stable generators.

Power Boundedness and Kreiss Conditions

The paper demonstrates that the ordinary Kreiss condition does not imply strong Kreiss boundedness or power boundedness for Cayley transforms of semigroup generators.

For any $\alpha \in (0, 1)$ , there exist generators where the Cayley transform $C_h(A)$ satisfies the ordinary Kreiss condition with a uniform constant but is not strongly Kreiss bounded.
The powers of the Cayley transform satisfy a lower bound:
$\|C_h(A)^m\| \gtrsim (\log \log(m/h + e^e))^\alpha$
This resolves an open question regarding the optimality of the logarithmic growth factor in general Hilbert space estimates.

Stability of the Crank–Nicolson Scheme

Consequently, the Crank–Nicolson scheme is shown to be unstable in the operator norm for general bounded generators on Hilbert spaces, even for exponentially stable semigroups.

Long-time instability: For any fixed step size $h$ , the approximation error grows unboundedly as time $t \to \infty$ .
Mesh refinement instability: For any fixed final time $T$ , the approximation error grows unboundedly as the step size $h \to 0$ (i.e., as the number of steps $m \to \infty$ ).
The divergence is of order $(\log \log m)^\alpha$ .

4. Significance and Claims

The authors claim that their results provide definitive negative answers to the three questions listed above.

Optimality: The dimension-dependent separation between the ordinary and strong Kreiss constants is shown to be essentially optimal (up to an arbitrarily small power loss) due to the exponent $\alpha < 1$ .
Contrast with Previous Results: The work highlights that previous positive results on Hilbert spaces relied on specific structural assumptions (contractivity, analyticity, or sectoriality) which are not satisfied by the constructed counterexamples.
Numerical Implications: The instability of the Crank–Nicolson scheme implies that for general $C_0$ -semigroups on Hilbert spaces, convergence cannot be guaranteed solely based on the boundedness of the generator; regularity of the initial data is required for convergence, as the scheme may fail for certain initial vectors in the space.

The paper concludes that the "doubly logarithmic" growth is a fundamental feature of these counterexamples, arising from the specific interplay between the doubly exponential decay of eigenvalues and the unconditionality constants of the underlying bases.