Noncommutative complex analysis and Bargmann-Segal multipliers

We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on which satisfy them. These operators arise from the creation operator via a functional calculus involving a class of entire functions, identified by Newman and Shapiro, which act as unbounded multiplication operators on Bargmann-Segal space.

     

On the Noncommutative Residue for Pseudodifferential Operators with log-Polyhomogeneous Symbols

We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion

Witt’s Theorem for Noncommutative Conic Curves

Let k be a field. We extend the main result in Nyman (J. Algebra 434, 90–114, 2015) to show that all homogeneous noncommutative curves of genus zero over k are noncommutative \(\mathbb {P}^{1}\)-bundles over a (possibly) noncommutative base. Using this result, we compute complete isomorphism invariants of homogeneous noncommutative curves of genus zero, allowing us to generalize a theorem of Witt.     

Decomposition of an order isomorphism between matrix-ordered Hilbert spaces

The purpose of this note is to show that any order isomorphism between noncommutative -spaces associated with von Neumann algebras is decomposed into a sum of a completely positive map and a completely co-positive map. The result is an version of a theorem of Kadison for a Jordan isomorphism on operator algebras.

     

Hartree–Fock Theory for Pseudorelativistic Atoms

We study the Hartree–Fock model for pseudorelativistic atoms, that is, atoms where the kinetic energy of the electrons is given by the pseudo-relativistic operator . We prove the existence of a Hartree–Fock minimizer, and prove regularity away from the nucleus and pointwise exponential decay of the corresponding orbitals. Submitted: August 1, 2007. Accepted: November 8, 2007.     

Almost global existence for quasilinear wave equations in three space dimensions

We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only use the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the decay of solutions of the wave equation as much as the decay. Accordingly, a key step in our approach is to prove a pointwise estimate of solutions of the wave equation that gives decay of solutions of the inhomogeneous linear wave equation in terms of a -weighted norm on the forcing term. A weighted space-time estimate for inhomogeneous wave equations is also important in making the spatial decay useful for the long-term existence argument.

     

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be aσ-finite von Neumann algebra and let AM be a maximal subdiagonal algebra with respect to a faithful normal conditional expectationΦ.Based on the Haagerup’s noncommutative Lpspace Lp(M)associated with M,we consider Toeplitz operators and the Hilbert transform associated with A.We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2(M)is just the right analytic Toeplitz algebra.Furthermore,the Hilbert transform on noncommutative Lp(M)is shown to be bounded for 1p∞.As an application,we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative H1(M)as a concrete space of operators.     

Eigenvalue asymptotics and exponential decay of eigenfunctions for Schrödinger operators with magnetic fields

We consider the Schrödinger operator with magnetic field,

Assuming that and is locally in certain reverse Hölder class, we study the eigenvalue asymptotics and exponential decay of eigenfunctions.

     

A noncommutative moment problem

We prove a noncommutative moment theorem and relate it to Connes' problem of embedding finite factor von Neumann algebras into an ultraproduct of the hyperfinite factor. We include a linear-algebraic equivalent of Connes' problem, which asks for a characterization of all noncommutative polynomials which have positive trace when the variables are replaced by contractive hermitian matrices.

     

Operator Hilbert spaces without the operator approximation property

We use a technique of Szankowski to construct operator Hilbert spaces that do not have the operator approximation property, including an example in a noncommutative space for .

     

Congruence preserving functions on free monoids

A function on an algebra is congruence preserving if for any congruence, it maps congruent elements to congruent elements. We show that on a free monoid generated by at least three letters, a function from the free monoid into itself is congruence preserving if and only if it is of the form \({x \mapsto w_{0}xw_{1} \cdots w_{n-1}xw_n }\) for some finite sequence of words \({w_0,\ldots ,w_n}\). We generalize this result to functions of arbitrary arity. This shows that a free monoid with at least three generators is a (noncommutative) affine complete algebra. As far as we know, it is the first (nontrivial) case of a noncommutative affine complete algebra.     

On interpolation of noncommutative symmetric Hardy spaces

We proved the noncommutative analogue of Calderón’s result for fully symmetric spaces \(E_1\) and \(E_2\) on (0, 1) and for a finite von Neumann algebra \({{\mathcal {M}}}\). We also proved the noncommutative symmetric Hardy space’s analogue of Calderón’s result for fully symmetric spaces and for finite subdiagonal subalgebras.     

An intersection multiplicity in terms of -modules

The main aim of this paper is to discuss the relation between Serre's intersection multiplicity and the Euler form. The Euler form is defined to be an alternating sum of the length of -modules and is used by Mori and Smith to develop intersection theory over noncommutative rings. We show that they differ by a sign and that this relation is closely related to Serre's vanishing theorem.

     

Imaginary powers of Laplace operators

We show that if is a second-order uniformly elliptic operator in divergence form on , then . We also prove that the upper bounds remain true for any operator with the finite speed propagation property.

     

A -functional and the rate of convergence of some linear polynomial operators

We show that the -functional

where , is equivalent to the rate of convergence of a certain linear polynomial operator. This operator stems from a Riesz-type summability process of expansion by Legendre polynomials. We use the operator above to obtain a linear polynomial approximation operator with a rate comparable to that of the best polynomial approximation.

     

Stability analysis of abstract systems of Timoshenko type

We consider an abstract system of Timoshenko type

$$\begin{aligned} \left\{ {\begin{array}{l} \rho_1{{\ddot \varphi}} + a A^{\frac12}(A^{\frac12}\varphi + \psi) =0\\\rho_2{{\ddot \psi}} + b A \psi + a (A^{\frac12}\varphi + \psi) -\delta A^\gamma {\theta} = 0\\\rho_3{{\dot \theta}} + c A\theta + \delta A^\gamma {{\dot \psi}} =0 \end{array}} \right. \end{aligned}$$

where the operator \({A}\) is strictly positive selfadjoint. For any fixed \({\gamma \in {\mathbb{R}}}\), the stability properties of the related solution semigroup \({S(t)}\) are discussed. In particular, a general technique is introduced in order to prove the lack of exponential decay of \({S(t)}\) when the spectrum of the leading operator \({A}\) does not consist of eigenvalues only.

     

On the uniform Kreiss resolvent condition

Let B be a Banach space with norm ‖ · ‖ and identity operator I. We prove that, for a bounded linear operator T in B, the strong Kreiss resolvent condition

$\parallel (T - \lambda I)^{ - k} \parallel \leqslant \frac{M}{{(|\lambda | - 1)^k }}, |\lambda | > 1,k = 1,2, \ldots ,$

implies the uniform Kreiss resolvent condition

$\left\| {\sum\limits_{k = 0}^n {\frac{{T^k }}{{\lambda ^{k + 1} }}} } \right\| \leqslant \frac{L}{{|\lambda | - 1}}, |\lambda | > 1, n = 0,1,2, \ldots .$

We establish that an operator T satisfies the uniform Kreiss resolvent condition if and only if so does the operator T ^m for each integer m ? 2.

     

Some remarks on the operator of Foias and Williams

In this paper we study the Foias-Williams operator

where , and is a Hankel operator with symbol . We exhibit a relationship between the similarity of to a contraction and the rate of decay of , the absolute values of the Fourier coefficients of the symbol .

     

On a desingularization of the moduli space of noncommutative tori

It is shown that the moduli space of the noncommutative tori admits a natural desingularization by the group . Namely, we prove that the moduli space of pairs is homeomorphic to a punctured two-dimensional sphere. The proof is based on a correspondence (a covariant functor) between the complex and noncommutative tori.

     

Porous Medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)~(-σ)u) = f, 0 σ 1/2.This paper poses the problem over {t ∈ R~+, x ∈ R~n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.