On totally *-paranormal operators

In this paper we study some properties of a totally *-paranormal operator (defined below) on Hilbert space. In particular, we characterize a totally *-paranormal operator. Also we show that Weyl’s theorem and the spectral mapping theorem hold for totally *-paranormal operators through the local spectral theory. Finally, we show that every totally *-paranormal operator satisfies an analogue of the single valued extension property for W ²(D, H) and some of totally *-paranormal operators have scalar extensions.     

Counting Closed Orbits of Hyperbolic Diffeomorphisms

In this paper we count closed orbits of a hyperbolic diffeomorphism restricted to a basic set. In fact we shall be dealing with two types of extensions of a hyperbolic diffeomorphism: a finite group extension (also called G-extensions) and also an automorphism extension (also called (G, ρ)-extensions). In particular we present Chebotarev type theorems and prime orbit theorem for such diffeomorphisms. These counting results are in the form of an asymptotic formula derived in complete analogy with the number theoretic result. The main difficulty is extending the domain of analyticity of the zeta and L-functions and this is overcome by resorting to symbolic dynamics. Unlike the case of a flow, the proof of this extension result does not rely on the properties of the Ruelle–Perron–Frobenius operator. Also the counting results do not use any tauberian theorems. Received: January 7, 2005.     

RECOVERING SINGULARITIES FROM BACKSCATTERING IN TWO DIMENSIONS

We have shown that in two dimensions the leading singularities of the quantum mechanical scattering potential are determined by the backscattering data. We assume that the short range potential belongs to a suitable weighted Sobolev space, and by estimating the iterative terms in the Born-expansion we are able to show, that for example for Heaviside-type singularities across a smooth hypersurface, both the location and the size of the jump are recovered from backscattering.

The main part of the proof consists in getting sharp enough estimates for the first non-linear Born-term. These estimates are proven using a recent characterization of W ^1,p -functions due to P. Hajlasz, and a modification of the classical Triebel's Maximal Inequality.     

Local compactness for linear elasticity in irregular domains

For the theory of boundary value problems in linear elasticity, it is of crucial importance that the space of vector-valued L²-functions whose symmetrized Jacobians are square-integrable should be compactly embedded in L². For regions with the cone property this is usually achieved by combining Korn's inequalities and Rellich's selection theorem. We shall show that in a class of less regular regions Korn's second inequality fails whereas the desired compact embedding still holds true.     

The bernsteim polynomial in one variable

The lemma on b-functions is a result due to I.N.Bernstein about the existence of certain differential operators with polynomial coefficients.In this paper we give an elementary and constructive proof of this result that works well in one variable.Our method results in a simple formula for the Bernstein polynomial b(λ)and a recursive definition for a differential operator d(λ)that produces b(λ).As an application we consider two consequences about the poles of certain meromorphic functions defined by the analytic continuation of distributions.     

Spectral and Scattering Theory for teh Linear Boltzmann Equation in Exterior Domain

It is well-known that functions u ? W^m,p (Ω) can be extended by a bounded linear operator E to functions Eu ≦ W^m,p( R ⁿ), if Ω is C^M-regular and m ≦ M. Here we prove a corresponding result for grid-functions with extension operators E_h converging to E and mention some applications.     

An Answer to S. Simons’ Question on the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Operator and a Normal Cone Operator

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar’s constraint qualification—that is, whether or not “the sum theorem” is true—is the most famous open problem in Monotone Operator Theory. In his 2008 monograph “From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons’ question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.      

A two-dimensional Gauss-Kuzmin theorem for singular continued fractions

A very simple proof of a generalization of the Gauss-Kuzmin theorem for singular continued fractions is given by considering the transition operator defined in [Se] as an operator on the Banach space BV(W) of complex-valued functions of bounded variation on W = [0, ( )]. The upper bound obtained here implies that the convergence rate, O(αⁿ), with 0.17 ≤ α ≤ 0.47 < g, is better than that obtained in [DK].     

A Morita theorem for dual operator algebras

We prove that two dual operator algebras are weak^∗ Morita equivalent in the sense of [D.P. Blecher, U. Kashyap, Morita equivalence of dual operator algebras, J. Pure Appl. Algebra 212 (2008) 2401-2412] if and only if they have equivalent categories of dual operator modules via completely contractive functors which are also weak^∗-continuous on appropriate morphism spaces. Moreover, in a fashion similar to the operator algebra case, we characterize such functors as the module normal Haagerup tensor product with an appropriate weak^∗ Morita equivalence bimodule. We also develop the theory of the W^∗-dilation, which connects the non-selfadjoint dual operator algebra with the W^∗-algebraic framework. In the case of weak^∗ Morita equivalence, this W^∗-dilation is a W^∗-module over a von Neumann algebra generated by the non-selfadjoint dual operator algebra. The theory of the W^∗-dilation is a key part of the proof of our main theorem.     

Q-functions of Hermitian Contractions of Krein-Ovcharenko Type

In this paper operator-valued Q-functions of Krein-Ovcharenko type are introduced. Such functions arise from the extension theory of Hermitian contractive operators A in a Hilbert space ℌ. The definition is related to the investigations of M.G. Krein and I.E. Ovcharenko of the so-called Q_μ- and Q_M-functions. It turns out that their characterizations of such functions hold true only in the matrix valued case. The present paper extends the corresponding properties for wider classes of selfadjoint contractive extensions of A. For this purpose some peculiar but fundamental properties on the behaviour of operator ranges of positive operators will be used. Also proper characterizations for Q_μ- and Q_M-functions in the general operator-valued case are given. Shorted operators and parallel sums of positive operators will be needed to give a geometric understanding of the function-theoretic properties of the corresponding Q-functions.     

Functional determinants for general self-adjoint extensions of Laplace-type operators resulting from the generalized cone

In this article we consider the zeta regularized determinant of Laplace-type operators on the generalized cone. For arbitrary self-adjoint extensions of a matrix of singular ordinary differential operators modelled on the generalized cone, a closed expression for the determinant is given. The result involves a determinant of an endomorphism of a finite-dimensional vector space, the endomorphism encoding the self-adjoint extension chosen. For particular examples, like the Friedrich’s extension, the answer is easily extracted from the general result. In combination with (Bordag et al. in Commun. Math. Phys. 182(2):371–393, 1996), a closed expression for the determinant of an arbitrary self-adjoint extension of the full Laplace-type operator on the generalized cone can be obtained.     

p-Quasihyponormal operators have scalar extensions of order 6

In this paper we show that every p-quasihyponormal operator has a scalar extension of order 6, i.e., is similar to the restriction to a closed invariant subspace of a scalar operator of order 6, where 0<p<1. As a corollary, we get that every p-quasihyponormal operator with rich spectra has a nontrivial invariant subspace. Also we show that Aluthge transforms preserve an analogue of the single-valued extension property for W²(D,H) and an operator T.     

Extensions of bitriangular operators

We prove that every one-dimensional extension of a bitriangular operator has a cyclic commutant. We also prove that ifT is an extension of a bitriangular operator by an algebraic operator, then the weakly closed algebraW(T) generated byT has a separating vector.This work was partially supported by NSF Grant DMS-9401544.Participant, Workshop in Linear Analysis and Probability, Texas A&M University     

Vector Lattices Associated with Ordered Vector Spaces

Let V be a real, Archimedian ordered, vector space, whose positive cone V ⁺ satisfies V =  V ⁺ – V ⁺. To V we associate a Dedekind complete vector lattice W containing V (by abuse of notation). In the case when V has an order unit the determination of W is already known. Let W₀ ì W{W_0 \subset W} be the vector lattice generated by V. We study W ₀ in the case when the cone C of all positive linear forms on V separates the elements of V. The determination of W ₀ involves the extreme rays of C. We determine the cone of positive linear forms on W ₀ in terms of conical measures on C.     

Product type bounds on the approximation of values ofE andG functions

This paper obtains effective lower bounds on the absolute values of linear forms, over the integers, in power products of values of certain SiegelE-functions or SiegelG-functions. The bounds obtained are in terms of the product of the absolute values of the coefficients. ForE-functions the bound obtained is a best possible result, up to an arbitrarily small positive epsilon. ForG-functions the result is asymptotically best in the following sense: for each epsilon larger than zero there exists an integerN such that ifz, the point of evaluation, equalsM ^–1 whereM is an integer with absolute value larger thanN, then the bound obtained is within epsilon of a best possible bound. (From the proof it is clear thatz need not be the inverse of an integer. What is necessary that the absolute value of its numerator must be much smaller than the absolute value of its denominator.)Results obtained recently by D. V. andG. V. Chudnovsky bounding the absolute values of similar forms give bounds in terms of the maximum of the absolute values of the coefficients; such lower bounds can be much smaller.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

L p bounds for a maximal dyadic sum operator

The authors prove L ^p bounds in the range 1<p< for a maximal dyadic sum operator on R ⁿ . This maximal operator provides a discrete multidimensional model of Carlesons operator. Its boundedness is obtained by a simple twist of the proof of Carlesons theorem given by Lacey and Thiele [7] adapted in higher dimensions [9]. In dimension one, the L ^p boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunts extension [4] of Carlesons theorem on almost everywhere convergence of Fourier integrals. Mathematics Subject Classification (2000):Primary 42A20, Secondary 42A24Grafakos is supported by the NSF. Tao is a Clay Prize Fellow and is supported by a grant from the Packard Foundation.     

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We give a new proof and a partial generalization of Jean Taylor’s result (Ann. Math. (2) 103(3), 489–539, 1976) that says that Almgren almost-minimal sets of dimension 2 in ℝ³ are locally C ^1+α -equivalent to minimal cones. The proof is rather elementary, but uses a local separation result proved in Ann. Fac. Sci. Toulouse 18(1), 65–246, 2009 and an extension of Reifenberg’s parameterization theorem (David et al. in Geom. Funct. Anal. 18, 1168–1235, 2008). The key idea is still that if X is the cone over an arc of small Lipschitz graph in the unit sphere, but X is not contained in a disk, we can use the graph of a harmonic function to deform X and substantially diminish its area. The local separation result is used to reduce to unions of cones over arcs of Lipschitz graphs. A good part of the proof extends to minimal sets of dimension 2 in ℝ ⁿ , but in this setting our final regularity result on E may depend on the list of minimal cones obtained as blow-up limits of E at a point.