Problems

Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent…

Iterates of a firmly nonexpansive operator converge weakly but not strongly, by Genel and Lindenstrauss. Whether their Cesaro means converge strongly was open. They need not: an explicit curve gives a counterexample.

The Howland-Kato conjecture that every nonzero positive commutator i[f(P),g(Q)] must arise from functions in appropriate Kato classes is false: i[arctan(P),arctan(Q)] is nonzero and nonnegative.

For p ≥ 2, does Carbery's proposed many-function almost-orthogonality inequality hold with the pairwise overlap coefficients raised to the power 2 - and if not, what is the largest possible exponent?

Aldroubi, Cabrelli, Krishtal and Molter conjectured that for a bounded normal operator T and any vector g, the normalized orbit T^k g / |T^k g| : k ≥ 0 is never a frame. It can be: an explicit construction produces a normalized orbit that…

Can one hear the shape of a drum, in the Steklov setting and in the plane? No: there exist pairs of noncongruent bounded plane domains with identical Steklov spectra including multiplicities, simply connected, strictly convex, with…

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…

A countable discrete group with a proper length function carries a natural spectral triple on its reduced group C*-algebra. A well-studied question in non-commutative metric geometry asks whether the associated Connes pseudo-metric always…

The Smith-Ward theorem realizes the first k essential matrix ranges of an operator as the matrix ranges of a compact perturbation. The classical Smith-Ward problem asks whether that perturbation can be chosen independently of k,…

Let μ be a probability measure on the unit circle with Verblunsky coefficients α. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of α into components localized at…

The unrestricted planar Berenstein conjecture holds that overdetermined Dirichlet-Neumann data characterize the disc. Disproved: a bounded simply connected domain with real-analytic Jordan boundary that is not a disc, carrying a nonzero…

Does every lattice of density above one admit a Gabor frame with a nice window? No. For every dimension d > 1 there are explicit criteria on lattices Λ ⊂ R^2d with D(Λ) > 1 such that no function with continuous Zak transform generates a…