Prescribed diagonal Schouten tensor in locally conformally flat manifolds

We consider the pseudo-euclidean space ${(\mathbb{R}^n, g)}$ , with n ≥  3 and ${g_{ij} = \delta_{ij} \varepsilon_i, \varepsilon_i = \pm 1}$ and tensors of the form ${T = \sum \nolimits_i \varepsilon_i f_i (x) dx_i^2}$ . In this paper, we obtain necessary and sufficient conditions for a diagonal tensor to admit a metric ${\bar{g}}$ , conformal to g, so that ${A_{\bar g}=T}$ , where ${A_{\bar g}}$ is the Schouten Tensor of the metric ${\bar g}$ . The solution to this problem is given explicitly for special cases for the tensor T, including a case where the metric ${\bar g}$ is complete on ${\mathbb{R}^n}$ . Similar problems are considered for locally conformally flat manifolds. As an application of these results we consider the problem of finding metrics ${\bar g}$ , conformal to g, such that ${\sigma_2 ({\bar g })}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })}}$ is equal to a given function. We prove that for some functions, f ₁ and f ₂, there exist complete metrics ${\bar{g} = g/{\varphi^2}}$ , such that ${\sigma_2 ({\bar g }) = f_1}$ or ${\frac{\sigma_2 ({\bar g })}{\sigma_1 ({\bar g })} = f_2}$ .     

A Geometry for the Set of Split Operators

We study the set ${\mathcal{X}}$ of split operators acting in the Hilbert space ${\mathcal{H}}$ : $$\mathcal{X}=\{T\in \mathcal{B}(\mathcal{H}): N(T)\cap R(T)=\{0\} \ {\rm and} \ N(T)+R(T)=\mathcal{H}\}.$$ Inside ${\mathcal{X}}$ , we consider the set ${\mathcal{Y}}$ : $$\mathcal{Y}=\{T\in\mathcal{X}: N(T)\perp R(T)\}.$$ Several characterizations of these sets are given. For instance ${T\in\mathcal{X}}$ if and only if there exists an oblique projection ${Q}$ whose range is N(T) such that T + Q is invertible, if and only if T posseses a commuting (necessarilly unique) pseudo-inverse S (i.e. TS = ST, TST = T and STS = S). Analogous characterizations are given for ${\mathcal{Y}}$ . Two natural maps are considered: $${\bf q}:\mathcal{X} \to \mathbb{Q}:=\{{\rm oblique \ projections \ in} \, \mathcal{H} \}, \ {\bf q}(T)=P_{R(T)//N(T)}$$ and $${\bf p}:\mathcal{Y} \to \mathbb{P}:=\{{\rm orthogonal \ projections \ in} \ \mathcal{H} \}, \ {\bf p}(T)=P_{R(T)}, $$ where ${P_{R(T)//N(T)}}$ denotes the projection onto R(T) with nullspace N(T), and P _R(T) denotes the orthogonal projection onto R(T). These maps are in general non continuous, subsets of continuity are studied. For the map q these are: similarity orbits, and the subsets ${\mathcal{X}_{c_k}\subset \mathcal{X}}$ of operators with rank ${k<\infty}$ , and ${\mathcal{X}_{F_k}\subset\mathcal{X}}$ of Fredholm operators with nullity ${k<\infty}$ . For the map p there are analogous results. We show that the interior of ${\mathcal{X}}$ is ${\mathcal{X}_{F_0}\cup\mathcal{X}_{F_1}}$ , and that ${\mathcal{X}_{c_k}}$ and ${\mathcal{X}_{F_k}}$ are arc-wise connected differentiable manifolds.     

Trivectors yielding spreads in PG(5, 2)

Given an alternating trilinear form ${T\in {\rm Alt}(\times^{3}V_{6})}$ on V ₆ = V(6, 2) let ${\mathcal{L}_{T}}$ denote the set of those lines ${\langle a, b \rangle}$ in ${{\rm PG}(5,2)=\mathbb{P}V_{6}}$ which are T-singular, satisfying, that is, T(a, b, x) = 0 for all ${x\in {\rm PG}(5, 2).}$ If ${\mathcal{L}_{21}}$ is a Desarguesian line-spread in PG(5, 2) it is shown that ${\mathcal{L}_{T}=\mathcal{L}_{21}}$ for precisely three choices T ₁,T ₂,T ₃ of T, which moreover satisfy T ₁ + T ₂ + T ₃ = 0. For ${T\in\mathcal{T}:=\{T_{1},T_{2},T_{3}\}}$ the ${\mathcal{G}_{T}}$ -orbits of flats in PG(5, 2) are determined, where ${\mathcal{G}_{T}\cong {\rm SL}(3,4).2}$ denotes the stabilizer of T under the action of GL(6, 2). Further, for a representative U of each ${\mathcal{G}_{T}}$ -orbit, the T-associate U ^# is also determined, where by definition $$U^{\#}=\{v\in {\rm PG}(5,2)\, |\, T(u_{1},u_{2},v) = 0\, \,{\rm for\,all }\, \, u_{1},u_{2}\in U\}$$ .     

Fixed point homomorphisms for parameterized maps

Let X be an ANR (absolute neighborhood retract), ${\Lambda}$ a k-dimensional topological manifold with topological orientation ${\eta}$ , and ${f : D \rightarrow X}$ a locally compact map, where D is an open subset of ${X \times \Lambda}$ . We define Fix(f) as the set of points ${{(x, \lambda) \in D}}$ such that ${x = f(x, \lambda)}$ . For an open pair (U, V) in ${X \times \Lambda}$ such that ${{\rm Fix}(f) \cap U \backslash V}$ is compact we construct a homomorphism ${\Sigma_{(f,U,V)} : H^{k}(U, V ) \rightarrow R}$ in the singular cohomologies H* over a ring-with-unit R, in such a way that the properties of Solvability, Excision and Naturality, Homotopy Invariance, Additivity, Multiplicativity, Normalization, Orientation Invariance, Commutativity, Contraction, Topological Invariance, and Ring Naturality hold. In the case of a ${C^{\infty}}$ -manifold ${\Lambda}$ , these properties uniquely determine ${\Sigma}$ . By passing to the direct limit of ${\Sigma_{(f,U,V)}}$ with respect to the pairs (U, V) such that ${K = {\rm Fix}(f) \cap U \backslash V}$ , we define a homomorphism ${\sigma_{(f,K)} : {H}_{k}({\rm Fix}(f), Fix(f) \backslash K) \rightarrow R}$ in the ?ech cohomologies. Properties of ${\Sigma}$ and ${\sigma}$ are equivalent each to the other. We indicate how the homomorphisms generalize the fixed point index.     

Characterization of (generalized) Lie derivations on {\mathcal{J}}-subspace lattice algebras by local action

Let ${\mathcal{L}}$ be a ${\mathcal{J}}$ -subspace lattice on a Banach space X over the real or complex field ${\mathbb{F}}$ with dim X ≥ 2 and Alg ${\mathcal{L}}$ be the associated ${\mathcal{J}}$ -subspace lattice algebra. For any scalar ${\xi \in \mathbb{F}}$ , there is a characterization of any linear map L : Alg ${\mathcal{L} \rightarrow {\rm Alg} {\mathcal{L}}}$ satisfying ${L([A,B]_\xi) = [L(A),B]_\xi + [A,L(B)]_\xi}$ for any ${A, B \in{\rm Alg} {\mathcal{L}}}$ with AB = 0 (rep. ${[A,B]_ \xi = AB - \xi BA = 0}$ ) given. Based on these results, a complete characterization of (generalized) ξ-Lie derivations for all possible ξ on Alg ${\mathcal{L}}$ is obtained.     

On Deformations of Lie Algebroids

For any Lie algebroid A, its 1-jet bundle ${\mathfrak{J} A}$ is a Lie algebroid naturally and there is a representation ${\pi:\mathfrak{J} A\longrightarrow\mathfrak{D} A}$ . Denote by ${{\rm d}_{\mathfrak{J}}}$ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A,A)_{{\mathfrak{D}} A}),{\rm d}_{\mathfrak{J}})}$ of the cochain complex ${(\Gamma({\rm Hom}(\wedge^\bullet\mathfrak{J} A, A)),{\rm d}_\mathfrak{J})}$ .     

Eta Cocycles, Relative Pairings and the Godbillon–Vey Index Theorem

We prove a Godbillon?CVey index formula for longitudinal Dirac operators on a foliated bundle with boundary ${(X,\mathcal{F})}$ ; in particular, we define a Godbillon?CVey eta invariant on ${(\partial X,\mathcal{F}_{\partial}),}$ that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Moreover, employing the Godbillon?CVey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras, which in the present context takes the form ${0 \to \mathbf{\mathfrak{J}} \to \mathbf{\mathfrak{A}} \to \mathbf{\mathfrak{B}} \to 0}$ with ${ \mathbf{\mathfrak{J}}}$ dense and holomorphically closed in ${C^* (X,\mathcal{F})}$ and ${ \mathbf{\mathfrak{B}}}$ depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle ${(\tau_{GV}^r,\sigma_{GV})}$ for the pair ${\mathbf{\mathfrak{A}} \to \mathbf{\mathfrak{B}}}$ ; ${\tau_{GV}^r}$ is a cyclic cochain on ${\mathbf{\mathfrak{A}}}$ defined through a regularization à la Melrose of the usual Godbillon?CVey cyclic cocycle ?? _GV ; ?? _GV is a cyclic cocycle on ${\mathbf{\mathfrak{B}}}$ , obtained through a suspension procedure involving ?? _GV and a specific 1-cyclic cocycle (Roe??s 1-cocycle). We call ?? _GV the eta cocycle associated to ?? _GV . The Atiyah?CPatodi?CSinger formula is obtained by defining a relative index class ${{\rm Ind} (D,D^\partial) \in K_* (\mathbf{\mathfrak{A}}, \mathbf{\mathfrak{B}})}$ and establishing the equality ${\langle {\rm Ind} (D), [\tau_{GV}] \rangle\,=\,\langle {\rm Ind} (D,D^\partial), [(\tau^r_{GV}, \sigma_{GV})] \rangle}$ . The Godbillon?CVey eta invariant ?? _GV is obtained through the eta cocycle ?? _GV .     

Composition Operators on the Lipschitz Space of a Tree

The Lipschitz space ${\mathcal{L}}$ of an infinite tree T rooted at o is defined as the space consisting of the functions ${f : T \rightarrow \mathbb{C}}$ such that $$\beta_f = {\rm sup}\{|f(v) - f(v^-)| : v \in T\backslash\{o\}, \,v^- {\rm parent \, of \,} v\}$$ is finite. Under the norm ${\|f\|_\mathcal{L} = |f(o)|+\beta_f,\mathcal{L}}$ is a Banach space. In this article, the functions φ mapping T into itself whose induced composition operator ${C_{\varphi} : f \mapsto f \circ \varphi}$ on the Lipschitz space is bounded, compact, or an isometry, are characterized. Specifically, it is shown that the symbols of the bounded composition operators are the Lipschitz maps of T into itself viewed as a metric space under the edge-counting distance. The symbols inducing compact operators have finite range while those inducing isometries on ${\mathcal{L}}$ are precisely the onto maps fixing the root and whose images of neighboring vertices coincide or are themselves neighboring vertices. Finally, the spectrum of the operators ${C_\varphi}$ that are isometries is studied in detail.     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

Averaged Pointwise Bounds for Deformations of Schrödinger Eigenfunctions

Let (M,g) be an n-dimensional, compact Riemannian manifold and ${P_0(\hbar) = -\hbar{^2} \Delta_g + V(x)}$ be a semiclassical Schrödinger operator with ${\hbar \in (0,\hbar_0]}$ . Let ${E(\hbar) \in [E-o(1),E+o(1)]}$ and ${(\phi_{\hbar})_{\hbar \in (0,\hbar_0]}}$ be a family of L ²-normalized eigenfunctions of ${P_0(\hbar)}$ with ${P_0(\hbar) \phi_{\hbar} = E(\hbar) \phi_{\hbar}}$ . We consider magnetic deformations of ${P_0(\hbar)}$ of the form ${P_u(\hbar) = - \Delta_{\omega_u}(\hbar) + V(x)}$ , where ${\Delta_{\omega_u}(\hbar) = (\hbar d + i \omega_u(x))^*({\hbar}d + i \omega_u(x))}$ . Here, u is a k-dimensional parameter running over ${B^k(\epsilon)}$ (the ball of radius ${\epsilon}$ ), and the family of the magnetic potentials ${(w_u)_{u\in B^k(\epsilon)}}$ satisfies the admissibility condition given in Definition 1.1. This condition implies that k ≥ n and is generic under this assumption. Consider the corresponding family of deformations of ${(\phi_{\hbar})_{\hbar \in (0, \hbar_0]}}$ , given by ${(\phi^u_{\hbar})_{\hbar \in(0, \hbar_0]}}$ , where $$\phi_{\hbar}^{(u)}:= {\rm e}^{-it_0 P_u(\hbar)/\hbar}\phi_{\hbar}$$ for ${|t_0|\in (0,\epsilon)}$ ; the latter functions are themselves eigenfunctions of the ${\hbar}$ -elliptic operators ${Q_u(\hbar): ={\rm e}^{-it_0P_u(\hbar)/\hbar} P_0(\hbar) {\rm e}^{it_0 P_u(\hbar)/\hbar}}$ with eigenvalue ${E(\hbar)}$ and ${Q_0(\hbar) = P_{0}(\hbar)}$ . Our main result, Theorem1.2, states that for ${\epsilon >0 }$ small, there are constants ${C_j=C_j(M,V,\omega,\epsilon) > 0}$ with j = 1,2 such that $$C_{1}\leq \int\limits_{\mathcal{B}^k(\epsilon)} |\phi_{\hbar}^{(u)}(x)|^2 \, {\rm d}u \leq C_{2}$$ , uniformly for ${x \in M}$ and ${\hbar \in (0,h_0]}$ . We also give an application to eigenfunction restriction bounds in Theorem 1.3.     

Immunity properties and strong positive reducibilities

We use certain strong Q-reducibilities, and their corresponding strong positive reducibilities, to characterize the hyperimmune sets and the hyperhyperimmune sets: if A is any infinite set then A is hyperimmune (respectively, hyperhyperimmune) if and only if for every infinite subset B of A, one has ${\overline{K}\not\le_{\rm ss} B}$ (respectively, ${\overline{K}\not\le_{\overline{\rm s}} B}$ ): here ${\le_{\overline{\rm s}}}$ is the finite-branch version of s-reducibility, ??_ss is the computably bounded version of ${\le_{\overline{\rm s}}}$ , and ${\overline{K}}$ is the complement of the halting set. Restriction to ${\Sigma^0_2}$ sets provides a similar characterization of the ${\Sigma^0_2}$ hyperhyperimmune sets in terms of s-reducibility. We also show that no ${A \geq_{\overline{\rm s}}\overline{K}}$ is hyperhyperimmune. As a consequence, ${\deg_{\rm s}(\overline{K})}$ is hyperhyperimmune-free, showing that the hyperhyperimmune s-degrees are not upwards closed.     

Split Partial Isometries

A partial isometry V is said to be a split partial isometry if ${\mathcal{H}=R(V) + N(V)}$ , with R(V) ∩ N(V) = {0} (R(V) = range of V, N(V) = null-space of V). We study the topological properties of the set ${\mathcal{I}_0}$ of such partial isometries. Denote by ${\mathcal{I}}$ the set of all partial isometries of ${\mathcal{B}(\mathcal{H})}$ , and by ${\mathcal{I}_N}$ the set of normal partial isometries. Then $$\mathcal{I}_N\subset \mathcal{I}_0\subset \mathcal{I}, $$ and the inclusions are proper. It is known that ${\mathcal{I}}$ is a C ^∞-submanifold of ${\mathcal{B}(\mathcal{H})}$ . It is shown here that ${\mathcal{I}_0}$ is open in ${\mathcal{I}}$ , therefore is has also C ^∞-local structure. We characterize the set ${\mathcal{I}_0}$ , in terms of metric properties, existence of special pseudo-inverses, and a property of the spectrum and the resolvent of V. The connected components of ${\mathcal{I}_0}$ are characterized: ${V_0,V_1\in \mathcal{I}_0}$ lie in the same connected component if and only if $${\rm dim}\, R(V_0)= {\rm dim}\, R(V_1) \,\,{\rm and}\,\,\, {\rm dim}\, R(V_0)^\perp = {\rm dim}\, R(V_1)^\perp.$$ This result is known for normal partial isometries.     

A Note on Drazin Invertibility for Upper Triangular Block Operators

A bounded linear operator A acting on a Banach space X is said to be an upper triangular block operators of order n, and we write ${A \in \mathcal{UT}_{n}(X)}$ , if there exists a decomposition of ${X = X_{1} \oplus . . . \oplus X_{n}}$ and an n × n matrix operator ${(A_{i,j})_{\rm 1 \leq i, j \leq n}}$ such that ${A = (A_{i, j})_{1 \leq i, j \leq n}, A_{i, j} = 0}$ for i > j. In this note we characterize a large set of entries A _{i, j} with j > i such that ${\sigma_{\rm D} (A) = {\bigcup\limits_{i = 1}^{n}} \sigma_{\rm D} (A_{i, i})}$ ; where σ_D(.) is the Drazin spectrum. Some applications concerning the Fredholm theory and meromorphic operators are given.     

Closed Range Property for Holomorphic Semi-Fredholm Functions

Given Banach spaces X and Y, we show that, for each operator-valued analytic map ${\alpha \in \mathcal O (D,\mathcal L(Y,X))}$ satisfying the finiteness condition ${\dim (X/\alpha (z)Y) < \infty}$ pointwise on an open set D in ${\mathbb {C}^n}$ , the induced multiplication operator ${\mathcal O(U,Y) \stackrel{\alpha}{\longrightarrow} \mathcal O (U,X)}$ has closed range on each Stein open set ${U \subset D}$ . As an application we deduce that the generalized range ${{\rm R}^{\infty}(T) = \bigcap_{k \geq 1}\sum_{| \alpha | = k} T^{\alpha}X}$ of a commuting multioperator ${T \in \mathcal L(X)^n}$ with ${\dim(X/\sum_{i=1}^n T_iX) < \infty}$ can be represented as a suitable spectral subspace.     

On the pullback of an arithmetic theta function

In this paper, we describe a relationship between the simplest examples of arithmetic theta series. The first of these are the weight 1 theta series ${\widehat{\phi}_{\mathcal C}(\tau)}$ defined using arithmetic 0-cycles on the moduli space ${\mathcal C}$ of elliptic curves with CM by the ring of integers ${O_{\kappa}}$ of an imaginary quadratic field. The second such series ${\widehat{\phi}_{\mathcal M}(\tau)}$ has weight 3/2 and takes values in the arithmetic Chow group ${\widehat{{\rm CH}}^1(\mathcal{M})}$ of the arithmetic surface associated to an indefinite quaternion algebra ${B/\mathbb{Q}}$ . For an embedding ${O_\kappa \rightarrow O_B}$ , a maximal order in B, and a two sided O _B -ideal Λ, there is a morphism ${j_\Lambda:{\mathcal C} \rightarrow {\mathcal M}}$ and a pullback ${j_\Lambda^*: \widehat{{\rm CH}}^1(\mathcal{M}) \rightarrow \widehat{{\rm CH}}^1(\mathcal C)}$ . Our main result is an expression for the pullback ${j^*_\Lambda \widehat{\phi}_{\mathcal M}(\tau)}$ as a linear combination of products of ${\widehat{\phi}_{\mathcal C}(\tau)}$ ’s and classical weight ${\frac{1}{2}}$ theta series.