Norm estimates of almost Mathieu operators

We estimate the norm of the almost Mathieu operator , regarded as an element in the rotation C^*-algebra . In the process, we prove for every λ∈R and the inequality

     

Quasi sure analysis of local times of anticipating smooth semimartingales

Let be the anticipating smooth semimartingale and be its generalized local time. In this paper, we give some estimates about the quasi sure property of Xt and its quadratic variation process t〈X〉. We also study the fractional smoothness of and prove that the quadratic variation process of can be constructed as the quasi sure limit of the form , where is a sequence of subdivisions of [a,b], , i=0,1,…,n².     

Homological characterization of the unknot

Given a knot K in the 3-sphere, let Q_K be its fundamental quandle as introduced by Joyce. Its first homology group is easily seen to be . We prove that H₂(Q_K)=0 if and only if K is trivial, and whenever K is non-trivial. An analogous result holds for links, thus characterizing trivial components.More detailed information can be derived from the conjugation quandle: let Q_K^π be the conjugacy class of a meridian in the knot group . We show that , where p is the number of prime summands in a connected sum decomposition of K.     

Riccati inequality and functional properties of differential operators on the half line

Given a piecewise continuous function and a projection P₁ onto a subspace X₁ of CN, we investigate the injectivity, surjectivity and, more generally, the Fredholm properties of the ordinary differential operator with boundary condition . This operator acts from the “natural” space into L²×X₁. A main novelty is that it is not assumed that A is bounded or that has any dichotomy, except to discuss the impact of the results on this special case. We show that all the functional properties of interest, including the characterization of the Fredholm index, can be related to the existence of a selfadjoint solution H of the Riccati differential inequality . Special attention is given to the simple case when H=A+A^∗ satisfies this inequality. When H is known, all the other hypotheses and criteria are easily verifiable in most concrete problems.     

Global branch from the second eigenvalue for a semilinear Neumann problem in a ball

Let (n?3) be a ball, and let f∈C³. We are concerned with the Neumann problem

     

Scattering operator for nonlinear Klein-Gordon equations in higher space dimensions

We prove the existence of the scattering operator in the neighborhood of the origin in the weighted Sobolev space Hβ,1 with for the nonlinear Klein-Gordon equation with a power nonlinearity

     

Toeplitz operators associated to commuting row contractions

Let H be a complex Hilbert space and let {Tn}_n?1 be a sequence of commuting bounded operators on H such that . Let denote the space of all operators X in B(H) for which and suppose that . We will show that there exists a triple {K,Γ,{Un}_n?1} where K is a Hilbert space, Γ:K→H is a bounded operator and {Un}_n?1⊂B(K) is a sequence of commuting normal operators with such that TnΓ=ΓUn for n?1, and for which the mapping Y?ΓYΓ^∗ is a complete isometry from the commutant of {Un}_n?1 onto the space . Moreover we show that the inverse of this mapping can be extended to a ^∗-homomorphism

     

Duality techniques for certain integral transforms to be starlike

For β<1, let denote the class of all normalized analytic functions f such that

     

On a paper by Gesztesy, Simon, and Teschl concerning isospectral deformations of ordinary Schrödinger operators

Starting from a selfadjoint Schrödinger operator A=−d²/dx²+q with a gap G in its spectrum F. Gesztesy, B. Simon, G. Teschl [J. Analyse Math. 70 (1996) 267-324] succeed in constructing another Schrödinger operator that is unitarily equivalent (and thus isospectral) to A. As the means they apply come from the Weyl-Titchmarsh theory the connections prove to be intricate, in particular the relation between A and . We show that a central assertion in GST's paper rests substantially on factorizations of the form

     

On strong uniform distribution III

Let a = (a_ii=1^∞ be a strictly increasing sequence of natural numbers and let be a space of Lebesgue measurable functions defined on [0,1). Let <y> denote the fractional part of the real number y. We say that a is an ^∗ sequence if for each f ?

     

Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

Browder spectra of upper-triangular operator matrices

Let be a 2×2 upper triangular operator matrix acting on the Hilbert space H⊕K. In this paper, for given operators A and B, we prove that

     

Sharp estimates for the non-centered maximal operator associated to Gaussian and other radial measures

In this work we investigate the integrability properties of the maximal operator M_μ, associated with a non-doubling measure μ defined on the Euclidean space , with special emphasis on the Gaussian and similar measures. Among other results we show for a wide class of radial and decreasing measures μ, that M_μ satisfies the modular inequality

     

Second-order energies on thin structures: variational theory and non-local effects

For a given positive measure μ on , we consider integral functionals of the kind