P-adic discrete semigroup of contractions

Let A ∈ B(X) be a spectral operator on a non-archimedean Banach space over Cp. In this paper, we give a necessary and sufficient condition on the resolvent of A so that the discrete semigroup consisting of powers of A is contractions.

Spectral theories and topological strings on del Pezzo geometries

Motivated by understanding M2-branes, we propose to reformulate partition functions of M2-branes by quantum curves. Especially, we focus on the backgrounds of del Pezzo geometries, which enjoy Weyl group symmetries of exceptional algebras. We construct quantum curves explicitly and turn to the analysis of classical phase space areas and quantum mirror maps. We find that the group structure helps in clarifying previous subtleties, such as the shift of the chemical potential in the area and the identification of the overall factor of the spectral operator in the mirror map. We list the multiplicities characterizing the quantum mirror maps and find that the decoupling relation known for the BPS indices works for the mirror maps. As a result, with the group structure we can present explicitly the statement for the correspondence between spectral theories and topological strings on del Pezzo geometries.

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis

Given the abstract evolution equation y ′ ( t ) = A y ( t ) , t ∈ ℝ , y^{\prime} (t)=Ay(t),t\in {\mathbb{R}}, with a scalar type spectral operator A in a complex Banach space, we find conditions on A, formulated exclusively in terms of the location of its spectrum in the complex plane, necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1 \beta \ge 1 , in particular analytic or entire, on ℝ {\mathbb{R}} . We also reveal certain inherent smoothness improvement effects and show that if all weak solutions of the equation are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded. The important particular case of the equation with a normal operator A in a complex Hilbert space follows immediately.

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the open semi-axis

Given the abstract evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array}$$ with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly Gevrey ultradifferentiable of order β ≥ 1, in particular analytic or entire, on the open semi-axis (0, ∞). Also, revealed is a certain interesting inherent smoothness improvement effect.

On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator of orders less than one

It is shown that, if all weak solutions of the evolution equation $$\begin{array}{} \displaystyle y'(t)=Ay(t),\, t\ge 0, \end{array} $$ with a scalar type spectral operator A in a complex Banach space are Gevrey ultradifferentiable of orders less than one, then the operator A is necessarily bounded.

On the Differentiability of Weak Solutions of an Abstract Evolution Equation with a Scalar Type Spectral Operator on the Real Axis

Given the abstract evolution equation y′(t)=Ay(t), t∈R, with scalar type spectral operator A in a complex Banach space, found are conditions necessary and sufficient for all weak solutions of the equation, which a priori need not be strongly differentiable, to be strongly infinite differentiable on R. The important case of the equation with a normal operator A in a complex Hilbert space is obtained immediately as a particular case. Also, proved is the following inherent smoothness improvement effect explaining why the case of the strong finite differentiability of the weak solutions is superfluous: if every weak solution of the equation is strongly differentiable at 0, then all of them are strongly infinite differentiable on R.

Comment on “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator”

The results of three papers, in which the author inadvertently overlooks certain deficiencies in the descriptions of the Carleman classes of vectors, in particular the Gevrey classes, of a scalar type spectral operator in a complex Banach space established in “On the Carleman Classes of Vectors of a Scalar Type Spectral Operator,” Int. J. Math. Math. Sci. 2004 (2004), no. 60, 3219–3235, are observed to remain true due to more recent findings.