ProbXiv
sign in
unchecked

The Inverse Generator Problem on Hilbert Spaces

Nothing has been published against this problem here, and nobody has checked anything. That is the ordinary condition of an open problem, not a defect in the record.

a-counterexample-to-the-inverse-generator-problem-and-related-questionsAnalysisposed by Ralph deLaubenfels, 1988recorded: disproved

1 attempt · no person has looked

Statement

If AA generates a bounded C0C_0-semigroup on a Hilbert space and has dense range, does A−1A^{-1} also generate a bounded C0C_0-semigroup? Posed by deLaubenfels in 1988. Answered negatively: Lorist, Meyries and Veraar construct a bounded operator with dense range generating a bounded, strongly stable semigroup whose inverse generates no C0C_0-semigroup at all. The counterexamples come from one explicit finite-dimensional construction, using bases of C2n\mathbb{C}^{2n} with uniformly bounded partial-sum projections but unconditionality constants growing like nαn^\alpha.

Context

One finite-dimensional construction settles three related questions. Besides the inverse generator problem, it gives a generator whose Cayley transforms satisfy the ordinary Kreiss resolvent condition but are neither strongly Kreiss bounded nor power bounded, and it shows the Crank-Nicolson scheme is unstable in operator norm both over long times at fixed step size and under mesh refinement at fixed final time.

A named 1988 problem of semigroup theory with a 38-year ladder of partial results, motivated from numerical analysis, control theory and functional calculus, but read within those communities only.

People

no project yet · nobody looking

Projects

none yet

Nobody is running a project on this. A project is a stated goal, a thread, and one thing somebody else could do. It takes a title, one sentence on what would count as progress, and that one task.

begin a project on this problem →

Interest

nobody looking

Nobody has said they are looking at this. A mark here is a statement about you, not a claim on the problem: you set it, you clear it, and it blocks nobody.

Attempts

1 attempt

No person has examined this. 1 attempt is published here and nothing has been checked against it at all. Saying whether the mathematics holds is the most useful thing anybody can do on this page.

review this attempt →

  • #1

    Attempt 1

    constructionChatGPT 5.6 Pro, Claude Fable 5 with Emiel Lorist, Martin Meyries, Mark Veraar ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    ChatGPT 5.6 Pro, Claude Fable 5
    people
    Emiel Lorist, Martin Meyries, Mark Veraar

    The paper's disclosure in full: "ChatGPT 5.6 Pro by OpenAI was used to explore proof strategies and to check intermediate steps. Claude Fable 5 by Anthropic was used to check the arguments and detect mistakes. The authors take full responsibility for the content of this note." Strategy exploration and checking rather than an attributed step, so the lower tier applies.

    Reviews

    0 human reviews · 0 machine checks

    No person has reviewed this attempt. It has not been checked at all.

    Endorsements

    0 endorsements

    No one has endorsed this attempt. An endorsement is a person stating that they checked this version and believe it is correct. None has been recorded — which is information, not an omission.

    Discussion of this attempt

    no comments

Discussion

no comments

Nothing has been said about this problem yet. Discussion is for questions about the statement, pointers to prior work and objections to an attempt. It is not review: a review is a verdict recorded against one version of one attempt, and it is counted separately.

Reading every thread is open to everyone. Posting needs an account with posting rights — sign in to check yours.