Abstract We study the cyclicity of polynomials in Poletsky–Stessin weighted Bergman spaces on various domains in C 2 $\mathbb {C}^2$ double struck upper C squared , including the unit ball, the bidisk, and the complex ellipsoid. To this end, we introduce a natural extension of the parameter range for Poletsky–Stessin weighted Bergman spaces on complete Reinhardt domains, yielding a family of spaces that resemble Dirichlet-type spaces on the unit ball. We highlight the differences in the cyclicity behavior of polynomials in these spaces on the bidisk compared to those studied by Bénéteau et al. Finally, we propose several open problems concerning the structure of cyclic polynomials in these spaces.
Abstract In this article, we investigate several properties of the Volterra-type integral operator V g $V_g$ upper V Subscript g on the Fock space. We give an explicit description of its Berezin range, prove that this range is path connected, and establish several results concerning its convexity. We compute the operator norm, essential norm, Berezin norm, and Berezin number of V g $V_g$ upper V Subscript g . We also derive an exact coefficient representation of its numerical range and obtain a spectral characterization of its numerical radius. In particular, we prove that the operator norm, Berezin norm, Berezin number, and numerical radius coincide if and only if the polynomial expansion of the symbol g contains no linear term. This extremal identity holds even though the operator is non-normal.
We study the exactness of the reduced crossed product of a semigroup dynamical system and the reduced C ∗ $C^{*}$ upper C Superscript asterisk -algebra of a product system. We show that for a semigroup dynamical system ( A , P , α ) $(A, P,\alpha )$ left parenthesis upper A comma upper P comma alpha right parenthesis , under reasonable hypotheses (e.g., P is abelian and finitely generated), the reduced crossed product A ⋊ red P $A \rtimes _{\mathrm {red}} P$ upper A right normal factor semidirect product Subscript red Baseline upper P is exact if and only if A is exact. This strengthens our earlier result (Hossain and Sundar, 2026, J. Noncommut. Geom. ), where it was assumed that the action of P on A is by injective endomorphisms. We also compare the groupoid crossed product described in Hossain and Sundar (2026) and the Fell bundle constructed in Rennie et al. (2017, Math. Proc. Cambridge Philos. Soc. , 163, 561–580) for a product system, and show that they are equivalent as Fell bundles.
Abstract We are interested in an inverse problem for the fractional Schrödinger equation with a time-independent potential on closed Riemannian manifolds. The model is given by i ∂ t α u + ( ( − Δ g ) β + V ) u = f , $$\begin{align*}i\partial_t^\alpha u+\big((-\Delta_g)^\beta+V\big)u=f, \end{align*}$$ with α , β ∈ ( 0 , 1 ) $\alpha ,\beta \in (0,1)$ alpha comma beta element of left parenthesis 0 comma 1 right parenthesis and V ∈ L ∞ ( M ) $V\in L^\infty (M)$ upper V element of upper L Superscript infinity Baseline left parenthesis upper M right parenthesis real-valued. Our main goal is to determine the fractional powers α , β $\alpha ,\beta $ alpha comma beta , the Riemannian metric g (up to an isometry), and the potential V simultaneously from the knowledge of the associated source-to-solution map. For disjoint source and observation regions, the map determines α $\alpha $ alpha , β $\beta $ beta , and the complete cross-spectral data of ( − Δ g ) β + V $(-\Delta _g)^\beta +V$ left parenthesis minus normal upper Delta Subscript g Baseline right parenthesis Superscript beta Baseline plus upper V relative to these regions. For coinciding regions, if the potentials vanish there and the geometric and regularity hypotheses of Feizmohammadi et al. (2026, Invent. Math. ) hold, the map determines α $\alpha $ alpha , β $\beta $ beta , the metric (up to an isometry), and the potential simultaneously. Our argument builds on the work of Li (2026, Canad. Math. Bull. , 1–15) by combining the high-frequency behavior of the resolvent, Weyl’s law for the eigenvalues of the operator ( − Δ g ) β + V $(-\Delta _g)^\beta +V$ left parenthesis minus normal upper Delta Subscript g Baseline right parenthesis Superscript beta Baseline plus upper V , the unique continuation property of fractional operators with potentials, and the fractional Calderón theorem of Feizmohammadi et al. (2026). A probabilistic formulation of the inverse problem is also provided, and we prove that the corresponding uniqueness conclusions hold when the equation is driven by additive temporal white noise with a prescribed spatial profile.
Abstract In this note, we address a missing case in the proof of Proposition 5.3 from our article “Regularity and multiplicity of Veronese type algebras” (2026, Canad. Math. Bull. , 69, pp. 150–178) and rectify the statements of several subsequent results that depend on it.
