Neumann Theorem Research Articles

Focal points and sup-norms of eigenfunctions

If (M,g) is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order o(\lambda) saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point x_0 \in M for the geodesic flow at which the associated Perron–Frobenius operator U_{x_0}\colon L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M) has a nontrivial invariant L^2 function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.

Assignment problems with complementarities

The problem of allocating bundles of indivisible objects without transfers arises in many practical settings, including the assignment of courses to students, of siblings to schools, and of truckloads of food to food banks. In these settings, the complementarities in preferences are small compared with the size of the market. We exploit this to design mechanisms satisfying constrained efficiency and asymptotic strategy-proofness. We introduce two mechanisms, one for cardinal and the other for ordinal preferences. When agents do not want bundles of size larger than k, these mechanisms over-allocate each good by at most k−1 units, ex-post. These results are based on a generalization of the Birkhoff–von Neumann theorem on how probability shares of bundles can be expressed as lotteries over approximately feasible allocations, which is of independent interest.

Nonseparability and von Neumann's theorem for domains of unbounded operators

A classical theorem of von Neumann asserts that every unbounded self-adjoint operator $A$ in a separable Hilbert space $H$ is unitarily equivalent to an operator $B$ in $H$ such that $D(A)\cap D(B)=\{0\}$. Equivalently this can be formulated as a property for nonclosed operator ranges. We will show that von Neumann's theorem does not directly extend to the nonseparable case. In this paper we prove a characterisation of the property that an operator range $\mathcal{R}$ in a general Hilbert space $H$ admits a unitary operator $U$ such that $U\mathcal{R}\cap\mathcal{R}=\{0\}$. This allows us to study stability properties of operator ranges with the aforementioned property.

The Weyl–von Neumann theorem and Borel complexity of unitary equivalence modulo compacts of self-adjoint operators

The Weyl–von Neumann theorem asserts that two bounded self-adjoint operators A, B on a Hilbert space H are unitarily equivalent modulo compacts, i.e.uAu* + K = B for some unitary u 𝜖 u(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectrum: σess (A) = σess (B). We study, using methods from descriptive set theory, the problem of whether the above Weyl–von Neumann result can be extended to unbounded operators. We show that if H is separable infinite dimensional, the relation of unitary equivalence modulo compacts for bounded self-adjoint operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense Gδ-orbit but does not admit classification by countable structures. On the other hand, the apparently related equivalence relation A ~ B ⇔ ∃u 𝜖 U(H) [u(A-i)–1u* - (B-i)–1 is compact] is shown to be smooth.

Bounds for truncation errors of Graf’s and Neumann’s addition theorems

Graf's and Neumann's addition theorems for Bessel functions have been widely used in acoustic and electromagnetic scattering problems, especially the fast multipole method for 2-D scattering problems. This paper studies the truncation errors of Graf's and Neumann's addition theorems and their linear combinations. Explicit bounds and convergence rates of the truncation errors are derived, and convergence calculated. The conclusions are tested by numerical experiments and show that the derived bounds for the truncation errors of the addition theorems are valid.

The topology and geometry of self-adjoint and elliptic boundary conditions for Dirac and Laplace operators

The theory of self-adjoint extensions of first- and second-order elliptic differential operators on manifolds with boundary is studied via its most representative instances: Dirac and Laplace operators. The theory is developed by exploiting the geometrical structures attached to them and, by using an adapted Cayley transform on each case, the space [Formula: see text] of such extensions is shown to have a canonical group composition law structure. The obtained results are compared with von Neumann's theorem characterizing the self-adjoint extensions of densely defined symmetric operators on Hilbert spaces. The 1D case is thoroughly investigated. The geometry of the submanifold of elliptic self-adjoint extensions [Formula: see text] is studied and it is shown that it is a Lagrangian submanifold of the universal Grassmannian Gr. The topology of [Formula: see text] is also explored and it is shown that there is a canonical cycle whose dual is the Maslov class of the manifold. Such cycle, called the Cayley surface, plays a relevant role in the study of the phenomena of topology change. Self-adjoint extensions of Laplace operators are discussed in the path integral formalism, identifying a class of them for which both treatments leads to the same results. A theory of dissipative quantum systems is proposed based on this theory and a unitarization theorem for such class of dissipative systems is proved. The theory of self-adjoint extensions with symmetry of Dirac operators is also discussed and a reduction theorem for the self-adjoint elliptic Grassmannian is obtained. Finally, an interpretation of spontaneous symmetry breaking is offered from the point of view of the theory of self-adjoint extensions.

Von Neumann’s uniqueness theorem in theories with nonphysical particles

Von Neumann's uniqueness theorem is extended to a special class of canonical commutation relations, namely the anti-Fock representations, which are realized on a Krein space.

Realizations of rotations on an indecomposable compact monothetic group

By the classical Halmos‐von Neumann theorem, each compact monothetic group corresponds to an ergodic dynamical system with discrete spectrum. For such groups we prove two results. We first construct a compact monothetic group which does not split into a direct product of a connected and a totally disconnected compact monothetic group. Then we present a measure preserving dynamical system on the unit square being isomorphic to a rotation on this indecomposable group.

On Self-Adjoint Extensions and Symmetries in Quantum Mechanics

Given a unitary representation of a Lie group $G$ on a Hilbert space $\mathcal{H}$, we develop the theory of $G$-invariant self-adjoint extensions of symmetric operators both using von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the $G$-invariant unbounded operator. We also prove a $G$-invariant version of the representation theorem for quadratic forms. The previous results are applied to the study of $G$-invariant self-adjoint extensions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group $G$ acts. These extensions are labeled by admissible unitaries $U$ acting on the $L^2$-space at the boundary and having spectral gap at $-1$. It is shown that if the unitary representation $V$ of the symmetry group $G$ is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by $U$ is $G$-invariant if $U$ and $V$ commute at the boundary. Various significant examples are discussed at the end.

Categorification and Heisenberg doubles arising from towers of algebras

The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doubles and show that if induction and restriction satisfy Mackey-like isomorphisms then the Fock space representation of the Heisenberg double has a natural categorification. This unifies the existing categorifications of the polynomial representation of the Weyl algebra and the Fock space representation of the Heisenberg algebra. Second, we develop in detail the theory applied to the tower of 0-Hecke algebras, obtaining new Heisenberg-like algebras that we call quasi-Heisenberg algebras. As an application of a generalized Stone–von Neumann Theorem, we give a new proof of the fact that the ring of quasisymmetric functions is free over the ring of symmetric functions.

Exact and asymptotic results on coarse Ricci curvature of graphs

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we obtain the exact formulas for Ollivier’s Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature that hold for a wide class of graphs, and extend earlier results where the Ricci-curvature for graphs with girth 6 was obtained. We also prove a general lower bound on the Ricci-curvature in terms of the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize the Ricci-flat graphs of girth 5. Moreover, using our general lower bound and the Birkhoff–von Neumann theorem, we give the first necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci-curvature of random bipartite graphs G(n,n,p) and random graphs G(n,p), in various regimes of p.

Around the Van Daele–Schmüdgen Theorem

For a bounded non-negative self-adjoint operator acting in a complex, infinite-dimensional, separable Hilbert space \({\mathcal{H}}\) and possessing a dense range \({\mathcal{R}}\) we propose a new approach to characterisation of phenomenon concerning the existence of subspaces \({\mathfrak{M}\subset \mathcal{H}}\) such that \({\mathfrak{M}\cap\mathcal{R}=\mathfrak{M}^\perp\cap\mathcal{R}=\{0\}}\). We show how the existence of such subspaces leads to various pathological properties of unbounded self-adjoint operators related to von Neumann theorems (J Reine Angew Math 161:208–236, 1929; Math Ann 102:49–131, 1929; Math Ann 102:370–427, 1929). We revise the von Neumann–Van Daele-Schmüdgen assertions (J Reine Angew Math 161:208–236, 1929; J Oper Theory 11:379–393, 1984; Can J Math 36:1245–1250, 1982) to refine them. We also develop a new systematic approach, which allows to construct for any unbounded densely defined symmetric/self-adjoint operator T infinitely many pairs \({\langle T_1 , T_2 \rangle}\) of its closed densely defined restrictions \({T_k\subset T}\) such that \({{\rm dom}(T^* T_{k})=\{0\} (\Rightarrow{\rm dom} T_{k}^2=\{0\})\, k=1,2}\) and \({{\rm dom} T_1\cap{\rm dom} T_2=\{0\}, {\rm dom} T_1\dot+{\rm dom} T_2={\rm dom}T}\).

Birkhoff--von Neumann Theorem for Multistochastic Tensors

In this paper, we study the Birkhoff--von Neumann theorem for a class of multistochastic tensors. In particular, we give a necessary and sufficient condition such that a multistochastic tensor is a convex combination of finitely many permutation tensors. It is well-known that extreme points in the set of doubly stochastic matrices are just permutation matrices. However, we find that extreme points in the set of multistochastic tensors are not just permutation tensors. We provide the other types of tensors contained in the set of extreme points.

On the Vertices of the d-Dimensional Birkhoff Polytope

Let us denote by Ωn the Birkhoff polytope of n×n doubly stochastic matrices. As the Birkhoff---von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n×n permutation matrices. Here we consider a higher-dimensional analog of this basic fact. Let $\varOmega^{(2)}_{n}$ be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entries in which every line sums to 1. What can be said about $\varOmega ^{(2)}_{n}$'s vertex set? It is well known that an order-n Latin square may be viewed as a tristochastic array where every line contains nź1 zeros and a single 1 entry. Indeed, every Latin square of order n is a vertex of $\varOmega^{(2)}_{n}$, but as we show, such vertices constitute only a vanishingly small subset of $\varOmega^{(2)}_{n}$'s vertex set. More concretely, we show that the number of vertices of $\varOmega ^{(2)}_{n}$ is at least $(L_{n})^{\frac{3}{2}-o(1)}$, where Ln is the number of order-n Latin squares. We also briefly consider similar problems concerning the polytope of n×n×n arrays where the entries in every coordinate hyperplane sum to 1, improving a result from Kravtsov (Cybern. Syst. Anal., 43(1):25---33, 2007). Several open questions are presented as well.

An approximate version of the Jordan von Neumann theorem for finite-dimensional real normed spaces

It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite-dimensional normed vector spaces over , we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant for yields a small Banach–Mazur distance with , . Finally, we examine how this estimate worsens as increases, with the conclusion that grows quadratically with .

Amenability and uniqueness

The main result of this paper is a characterization of properly infinite injective von Neumann algebras and of nuclear C∗-algebras by using a uniqueness theorem, based on generalizations of Voiculescu’s famous Weyl–von Neumann theorem.

Convolution filters for polygons and the Petr–Douglas–Neumann theorem

We use the discrete Fourier transformation of planar polygons, convolution filters, and a shape function to give a very simple and enlightening description of circulant polygon transformations and their iterates. We consider in particular Napoleon’s and the Petr–Douglas–Neumann theorem.

An extension of the polytope of doubly stochastic matrices

We consider a class of matrices whose row and column sum vectors are majorized by given vectors b and c, and whose entries lie in the interval [0, 1]. This class generalizes the class of doubly stochastic matrices. We investigate the corresponding polytope Ω(b|c) of such matrices. Main results include a generalization of the Birkhoff–von Neumann theorem and a characterization of the faces, including edges, of Ω(b|c).

Abelian varieties and Weil representations

The main goal of this article is to construct and study a family of Weil representations over an arbitrary locally noetherian scheme without restriction on characteristic. The key point is to recast the classical theory in the scheme-theoretic setting. As in work of Mumford, Moret-Bailly and others, a Heisenberg group (scheme) and its representation can be naturally constructed from a pair of an abelian scheme and a nondegenerate line bundle, replacing the role of a symplectic vector space. Once enough is understood about the Heisenberg group and its representations (e.g., the analogue of the Stone–von Neumann theorem), it is not difficult to produce the Weil representation of a metaplectic group (functor) from them. As an interesting consequence (when the base scheme is SpecF¯p), we obtain the new notion of mod p Weil representations of p-adic metaplectic groups on F¯p-vector spaces. The mod p Weil representations admit an alternative construction starting from a p-divisible group with a symplectic pairing. We have been motivated by a few possible applications, including a conjectural mod p theta correspondence for p-adic reductive groups and a geometric approach to the (classical) theta correspondence.

On a Weyl–von Neumann type theorem for antilinear self-adjoint operators

Antilinear operators on a complex Hilbert space arise in various contexts in mathematical physics. In this paper, an analogue of the Weyl–von Neumann theorem for antilinear self-adjoint operators is proved, i.e. that an antilinear self-adjoint opera