The Inverse Generator Problem on Hilbert Spaces

If $A$ generates a bounded $C_0$-semigroup on a Hilbert space and has dense range, does $A^{-1}$ also generate a bounded $C_0$-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 $C_0$-semigroup at all. The counterexamples come from one explicit finite-dimensional construction, using bases of $\mathbb{C}^{2n}$ with uniformly bounded partial-sum projections but unconditionality constants growing like $n^\alpha$.

One finite-dimensional construction settles several 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. Version 2 adds Theorem 1.4, whose part (i) solves Question 6.1 of Chalmoukis, Tsikalas and Yakubovich; that question is tracked as its own entry.