Operator Ideals on Non-commutative Function Spaces

Suppose X and Y are Banach spaces, and \({{\mathcal{I}}}\) , \({{\mathcal{J}}}\) are operator ideals. compact operators). Under what conditions does the inclusion \({\mathcal{I}(X,Y) \subset \mathcal{J}(X,Y)}\) , or the equality \({\mathcal{I}(X,Y)\,=\,\mathcal{J}(X,Y)}\) , hold? We examine this question when \({\mathcal{I}, \mathcal{J}}\) are the ideals of Dunford–Pettis, strictly (co)singular, finitely strictly singular, inessential, or (weakly) compact operators, while X and Y are non-commutative function spaces. Since such spaces are ordered, we also address the same questions for positive parts of such ideals.     

Finite frames,P-frames and basically disconnected frames

Let \({\mathcal{P}}\) be an ideal of closed quotients of a completely regular frame L and \({\mathcal{R}_{\mathcal{P}}(L)}\) the collection of all functions in the ring \({\mathcal{R}(L)}\) whose support belong to \({\mathcal{P}}\) . We show that \({\mathcal{R}(L)}\) is a Noetherian ring if and only if \({\mathcal{R}(L)}\) is an Artinian ring if and only if L is a finite frame. Using this result, we next show that if \({\mathcal{P}}\) is the ideal of all compact closed quotients of L and L is \({\mathcal{P}}\) -continuous, then \({\mathcal{R}_{\mathcal{P}}(L)}\) is a Noetherian ring if and only if L is finite. Moreover, we show that L is a P-frame if and only if each ideal of \({\mathcal{R}(L)}\) is of the form \({\mathcal{R}_{\mathcal{P}}(L)}\) for some choice of \({\mathcal{P}}\) . We furnish equivalent conditions for \({\mathcal{R}_{\mathcal{P}}(L)}\) to be a prime ideal, a free ideal, and an essential ideal of \({\mathcal{R}(L)}\) separately in terms of the cozero elements of L. Finally, we show that L is basically disconnected if and only if \({\mathcal{R}(L)}\) is a coherent ring.     

Topological genericity of nowhere differentiable functions in the disc algebra

In this paper we introduce a class of functions contained in the disc algebra \({\mathcal{A}(D)}\) . We study functions \({f \in \mathcal{A}(D)}\) which have the property that the continuous periodic function \({u = {\rm Re}f|_{\mathbb{T}}}\) , where \({\mathbb{T}}\) is the unit circle, is nowhere differentiable. We prove that this class is non-empty and instead, generically, every function \({f \in \mathcal{A}(D)}\) has the above property. Afterwards, we strengthen this result by proving that, generically, for every function \({f \in \mathcal{A}(D)}\) , both continuous periodic functions \({u = {\rm Re}f|_\mathbb{T}}\) and \({\tilde{u} = {\rm Im}f|_\mathbb{T}}\) are nowhere differentiable. We avoid any use of the Weierstrass function and we mainly use Baire’s Category Theorem.     

Non-Veech surfaces in {\mathcal{H}^{\rm hyp}(4)} are generic

We show that every surface in the component \({\mathcal{H}^{\rm hyp}(4)}\) , that is the moduli space of pairs \({(M,\omega)}\) where M is a genus three hyperelliptic Riemann surface and \({\omega}\) is an Abelian differential having a single zero on M, is either a Veech surface or a generic surface, i.e. its \({{\rm GL}^{+}(2,\mathbb{R})}\) -orbit is either a closed or a dense subset of \({\mathcal{H}^{\rm hyp}(4)}\) . The proof develops new techniques applicable in general to the problem of classifying orbit closures, especially in low genus. Combined with work of Matheus and the second author, a corollary is that there are at most finitely many non-arithmetic Teichmüller curves (closed orbits of surfaces not covering the torus) in \({\mathcal{H}^{\rm hyp}(4)}\) .     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

Geometry of Normal Graphs in Euclidean Space and Applications to the Penrose Inequality in Minkowski

The Penrose inequality in Minkowski is a geometric inequality relating the total outer null expansion and the area of closed, connected and spacelike codimension-two surfaces \({{\bf \mathcal{S}}}\) in the Minkowski spacetime, subject to an additional convexity assumption. In a recent paper, Brendle and Wang A (Gibbons–Penrose inequality for surfaces in Schwarzschild Spacetime. arXiv:1303.1863, 2013) find a sufficient condition for the validity of this Penrose inequality in terms of the geometry of the orthogonal projection of \({{\bf \mathcal{S}}}\) onto a constant time hyperplane. In this work, we study the geometry of hypersurfaces in n-dimensional Euclidean space which are normal graphs over other surfaces and relate the intrinsic and extrinsic geometry of the graph with that of the base hypersurface. These results are used to rewrite Brendle and Wang’s condition explicitly in terms of the time height function of \({{\bf \mathcal{S}}}\) over a hyperplane and the geometry of the projection of \({{\bf \mathcal{S}}}\) along its past null cone onto this hyperplane. We also include, in Appendix, a self-contained summary of known and new results on the geometry of projections along the Killing direction of codimension two-spacelike surfaces in a strictly static spacetime.     

Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

Consider the nonlinear heat equation $$v_t -\Delta v=|v|^{p-1}v \qquad \qquad \qquad (NLH)$$ in the unit ball of \({\mathbb{R}^2}\) , with Dirichlet boundary condition. Let \({u_{p,\mathcal{K}}}\) be a radially symmetric, sign-changing stationary solution having a fixed number \({\mathcal{K}}\) of nodal regions. We prove that the solution of (NLH) with initial value \({\lambda u_{p,\mathcal{K}}}\) blows up in finite time if |λ ?1| > 0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of \({u_{p,\mathcal{K}}}\) and of the linearized operator \({L= -\Delta - p | u_{p,\mathcal{K}} | ^{p-1}}\) .     

The Cubic Complex Moment Problem

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\) . Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\) , then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\) . The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.     

Systems of variational inequalities for non-local operators related to optimal switching problems: existence and uniqueness

In this paper we study viscosity solutions to the system $$\begin{array}{ll} \min \{ -\mathcal{H}u_i(x,t)-\psi _i(x,t),u_i(x,t) - \max_{j \neq i} (-c_{i ,j} (x,t) + u_j (x,t)) \} = 0,\\ u_i(x,T)=g_i (x), \, i \in \{1,\ldots , d \},\end{array}$$ where \({(x,t)\in{\mathbb{R}}^{N} \times [0,T]}\) . Concerning \({{\mathcal{H}}}\) , we assume that \({{\mathcal{H}}={\mathcal{L}}+{\mathcal{I}}}\) where \({{\mathcal{L}}}\) is a linear, possibly degenerate, parabolic operator of second order and \({{\mathcal{I}}}\) is a non-local integro-partial differential operator. A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems when the dynamics of the underlying state variables is described by an N-dimensional Levy process. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth and structural assumptions on the data, i.e., on the operator \({{\mathcal{H}}}\) and on the continuous functions \({\psi_i}\) , c _i,j , and g _i . Using the comparison principle we prove the existence of a unique viscosity solution (u ₁, . . . , u _d ) to the system by Perron’s method. Our main contribution is that we establish existence and uniqueness of viscosity solutions, in the setting of Levy processes and non-local operators, with no sign assumption on the switching costs {c _{i, j} } and allowing c _{i, j} to depend on x as well as t.     

Irrational Factor of Order k and ITS Connections With k-Free Integers

We introduce an irrational factor of order k defined by \({I_{k}(n) ={\prod_{i=1}^{l}} p_{i}^{\beta_{i}}}\) , where \({n = \prod_{i=1}^{l} p_{i}^{\alpha_{i}}}\) is the factorization of n and \({\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i < k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}\) . It turns out that the function \({\frac{I_{k} (n)}{n}}\) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for \({\prod_{v=1}^{n} I_{k}(v)^{\frac{1}{n}}, \sum_{n \leqq x} I_{k}(n)}\) and \({\sum_{n \leqq x} (1 - \frac{n}{x}) I_{k}(n)}\) .     

Categorical group invariants of 3-manifolds

For a given class \({\mathcal{G}}\) of groups, a 3-manifold M is of \({\mathcal{G}}\) -category \({\leq k}\) if it can be covered by k open subsets such that for each path-component W of the subsets the image of its fundamental group \({ \pi_1(W) \rightarrow \pi(M )}\) belongs to \({\mathcal{G}}\) . The smallest number k such that M admits such a covering is the \({\mathcal{G}}\) -category, \({cat_{\mathcal{G}}(M)}\) . If M is closed, it has \({\mathcal{G}}\) -category between 1 and 4. We characterize all closed 3-manifolds of \({\mathcal{G}}\) -category 1, 2, and 3 for various classes \({\mathcal{G}}\) .     

A Jost–Pais-Type Reduction of Fredholm Determinants and Some Applications

We study the analog of semi-separable integral kernels in \({\mathcal {H}}\) of the type $$ K(x, x') = \left\{\begin{array}{ll} F_1(x) G_1(x'), \quad& a < x' < x < b,\\ F_2 (x)G_2(x'), \quad& a < x < x' < b,\end{array}\right.$$ where \({-\infty \leqslant a < b \leqslant \infty}\) , and for a.e. \({x \in (a, b)}\) , \({F_j (x) \in \mathcal{B}_2(\mathcal{H}_j, \mathcal{H})}\) and \({G_j(x) \in \mathcal {B}_2(\mathcal {H},\mathcal {H}_j)}\) such that F _j (·) and G _j (·) are uniformly measurable, and $$\begin{array}{ll} || F_j ( \cdot) ||_{\mathcal {B}_2(\mathcal {H}_j,\mathcal {H})} \in L^2((a, b)), ||G_j (\cdot)||_{\mathcal {B}_2(\mathcal {H},\mathcal {H}_j)} \in L^2((a, b)), \quad j=1,2, \end{array}$$ with \({\mathcal {H}}\) and \({\mathcal {H}_j}\) , j = 1, 2, complex, separable Hilbert spaces. Assuming that K(·, ·) generates a trace class operator K in \({L^2((a, b);\mathcal {H})}\) , we derive the analog of the Jost–Pais reduction theory that succeeds in proving that the Fredholm determinant \({{\rm det}_{L^2((a,b);\mathcal{H})}}\) (I ? α K), \({\alpha \in \mathbb{C}}\) , naturally reduces to appropriate Fredholm determinants in the Hilbert spaces \({\mathcal{H}}\) (and \({\mathcal{H}_1 \oplus \mathcal{H}_2}\) ). Explicit applications of this reduction theory to Schrödinger operators with suitable bounded operator-valued potentials are made. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.     

Supercritical Holes for the Doubling Map

For a map \({S : X \to X}\) and an open connected set (= a hole) \({H \subset X}\) we define \({\mathcal{J}_H(S)}\) to be the set of points in X whose S-orbit avoids H. We say that a hole H ₀ is supercritical if

1. for any hole H such that \({\overline{H}_0 \subset H}\) the set \({\mathcal{J}_H(S)}\) is either empty or contains only fixed points of S;

1. for any hole H such that \({\overline{H} \subset H_0}\) the Hausdorff dimension of \({\mathcal{J}_H(S)}\) is positive.

The purpose of this note is to completely characterize all supercritical holes for the doubling map Tx =  2x mod 1.     

Special values of anticyclotomic L-functions modulo λ

The purpose of this article is to generalize some results of Vatsal on the special values of Rankin–Selberg L-functions in an anticyclotomic \({\mathbb{Z}_{p}}\) -extension. Let g be a cuspidal Hilbert modular newform of parallel weight \({(2,\ldots,2)}\) and level \({\mathcal{N}}\) over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant \({\mathcal{D}}\) . We study the l-adic valuation of the special values \({L(g,\chi,\frac{1}{2})}\) as \({\chi}\) varies over the ring class characters of K of \({\mathcal{P}}\) -power conductor, for some fixed prime ideal \({\mathcal{P}}\) . We prove our results under the only assumption that the prime to \({\mathcal{P}}\) part of \({\mathcal{N}}\) is relatively prime to \({\mathcal{D}}\) .     

Diffusion by optimal transport in Heisenberg groups

We prove that the hypoelliptic diffusion of the Heisenberg group \({\mathbb{H }}_n\) describes, in the space of probability measures over \({\mathbb{H }}_n\) , a curve driven by the gradient flow of the Boltzmann entropy \({{\mathrm{Ent}}}\) , in the sense of optimal transport. We prove that conversely any gradient flow curve of \({{\mathrm{Ent}}}\) satisfy the hypoelliptic heat equation. This occurs in the subRiemannian \({\mathbb{H }}_n\) , which is not a space with a lower Ricci curvature bound in the metric sense of Lott–Villani and Sturm.     

Forbidden Triples Containing a Complete Graph and a Complete Bipartite Graph of Small Order

For a graph G and a set \({\mathcal{F}}\) of connected graphs, G is said be \({\mathcal{F}}\) -free if G does not contain any member of \({\mathcal{F}}\) as an induced subgraph. We let \({\mathcal{G} _{3}(\mathcal{F})}\) denote the set of all 3-connected \({\mathcal{F}}\) -free graphs. This paper is concerned with sets \({\mathcal{F}}\) of connected graphs such that \({\mathcal{F}}\) contains no star, \({|\mathcal{F}|=3}\) and \({\mathcal{G}_{3}(\mathcal{F})}\) is finite. Among other results, we show that for a connected graph T( ≠ K ₁) which is not a star, \({\mathcal{G}_{3}(\{K_{4},K_{2,2},T\})}\) is finite if and only if T is a path of order at most 6.