Modèles linéaires généralisés fonctionnels avec dérivée

We consider the functional generalized linear model whose response function is a linear operator depending on an explanatory variable X belonging to a functional space. It has been studied, among others, by Cardot and Sarda [4]. In this paper, we consider the functional generalized linear model with derivative component, denoted MLGFD in the following, whose response function depends on a linear operator of X and on its derivative. We propose estimators for the unknown functional parameters and provide convergence rates.     

On a Class of Analytic Operator Functions and Their Linearizations

We consider an operator function T in a Krein space which can formally be written as (0.1)but the last term on the right of (0.1) is replaced by a relatively form‐compact perturbation of a similar form. We study relations between the operator function T, a selfadjoint operator M in some Krein space, associated with T, and an operator which can be constructed with the help of the operator function –T^–1. The results are applied to a Sturm‐Liouville problem with a coefficient depending rationally on the eigenvalue parameter.     

On the eigenvalues of certain canonical higher-order ordinary differential operators

We consider the operator of taking the 2pth derivative of a function with zero boundary conditions for the function and its first p−1 derivatives at two distinct points. Our main result provides an asymptotic formula for the eigenvalues and resolves a question on the appearance of certain regular numbers in the eigenvalue sequences for p=1 and p=3.     

Riesz operators and the complete norm topology

We show that a non-injective Riesz operator on an infinite-dimensional Banach space X does not determine the complete norm topology of X. We also show that an injective operator with trivial generalized range determines the complete norm topology of X. Finally this result is used to settle the crucial role of the non-injectivity condition in our first result.     

Uniform estimates of Gronwall type

We discuss a Gronwall-type lemma depending on a small parameter ε>0, based on an integral inequality that predicts blow-up in finite time of the involved unknown function for any fixed ε. The result permits to establish uniform estimates even if the function itself depends on ε.     

Asymptotic type for sectorial operators and an integral of fractional powers

There is a standard notion of type for a sectorial linear operator acting in a Banach space. We introduce a notion of asymptotic type for a linear operator V, involving estimates on the resolvent ⁻¹(λI+V) as λ→0. We show, for example, that if V is sectorial and of asymptotic type ω then the fractional power Vα is of asymptotic type αω for a suitable range of positive α. Moreover, we establish various properties of the operator ; in particular, this operator is of asymptotic type 0, for a sectorial operator V. This result has an application to the construction of operators satisfying the well-known Ritt resolvent condition.     

Dependence of the Convergence Domain of Spectral Expansions on the Geometry of the Set of Discontinuity of the Function Being Expanded

For piecewise smooth functions, we indicate their domains of convergence and Riecz summarizability of their spectral expansions related to the Laplace operator in ? ⁿ , depending on the geometry of points of discontinuity of the function being expanded.     

On the functions of one selfadjoint integral operator

We construct a selfadjoint Akhiezer integral operator S on L ₂(?1, 1) with the spectrum {?1; 0; 1} and the property that every function φ(S) that is not a multiple of S fails to be an Akhiezer integral operator.     

Mixed modulation spaces and their application to pseudodifferential operators

This paper uses frame techniques to characterize the Schatten class properties of integral operators. The main result shows that if the coefficients {〈k,Φm,n〉} of certain frame expansions of the kernel k of an integral operator are in ?2,p, then the operator is Schatten p-class. As a corollary, we conclude that if the kernel or Kohn-Nirenberg symbol of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. Our corollary improves existing Schatten class results for pseudodifferential operators and the corollary is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class.     

Invariant subspaces of positive strictly singular operators on Banach lattices

It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators.     

On finite rank perturbations of definitizable operators

It was shown by P. Jonas and H. Langer that a selfadjoint definitizable operator A in a Krein space remains definitizable after a finite rank perturbation in resolvent sense if the perturbed operator B is selfadjoint and the resolvent set ρ(B) is nonempty. It is the aim of this note to prove a more general variant of this perturbation result where the assumption on ρ(B) is dropped. As an application a class of singular ordinary differential operators with indefinite weight functions is studied.     

The norm of a composition operator with linear symbol acting on the Dirichlet space

We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z)=az+b. We compare this result to an upper bound for ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.     

ω-hypoelliptic differential operators of constant strength

We study ω-hypoelliptic differential operators of constant strength. We show that any operator with constant strength and coefficients in which is homogeneous ω-hypoelliptic is also σ-hypoelliptic for any weight function σ=O(ω). We also present a sufficient condition in order to ensure that a differential operator admits a parametrix and, as a consequence, we obtain some conditions on the weights (ω,σ) to conclude that, for any operator P(x,D) with constant strength, the σ-hypoellipticity of the frozen operator P(x₀,D) implies the ω-hypoellipticity of P(x,D). This requires the use of pseudodifferential operators.     

Schrödinger operators with singular potentials from the space of multiplicators

We study Schrödinger operatorsT+Q, whereT=?Δ is the Laplace operator andQ is the multiplication operator by a generalized function (distribution). We also consider generalizations for the case of the polyharmonic operatorT = (-δ) ⁿ      

An Upper Bound of Essential Norm of Composition Operator on H2 （Bn）

The authors give an upper bound of the essential norm of a composition operator on H ²(B _n ), which involves the counting function in the higher dimensional value distribution theory defined by S. S. Chern. A criterion is also given to assure that the composition operator on H ²( _Bn ) is bounded or compact.     

AN ITERATIVE ALGORITHM FOR MAXIMAL MONOTONE MULTIVALUED OPERATOR EQUATIONS

1 IntroductionThe multivalued operator equations occur in various applications, e.g., mecha11ical systeimwith dry and viscous damping, electrical networks with switches, oscil1ations in viscoelastic-ity, optimization probIems with uonsmooth data, dynanilcal systems with nondifferentiablepotential, and optimal colltroI problellls. There have been a number of results, for instance,[1l-[6l, oll the solutions of multivallled operator equations. Amoug theln, R.T.Rockafellar[1]gave a prorimal poin…     

New Lorentz spaces for the restricted weak-type Hardy's inequalities

Associated to the class of restricted weak-type weights for the Hardy operator R_p, we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the classical Lorentz spaces. We also show that these new spaces are very natural to study the existence of equivalent norms described in terms of the maximal function.     

On multivariate Baskakov operator

In this paper, we study multivariate Baskakov operator Bn,d(f,x). We first show that the operator can retain some properties of the original function f, such as monotony, semi-additivity and Lipschitz condition, etc. Secondly, we discuss the monotony on the sequence of multivariate Baskakov operator Bn,d(f,x) for n when the function f is convex. Then, we propose, for estimating the rate of approximation, a new modulus of smoothness and prove the modulus to be equivalent to certain K-functional. Finally, with the modulus of smoothness as metric, we establish a strong direct theorem by using a decomposition technique for the operator.     

Existence of symmetric positive solutions for a class of Sturm-Liouville-like boundary value problems

This paper deals with the existence of symmetric positive solutions for a class of singular Sturm-Liouville-like boundary value problems with a one-dimensional p-Laplacian operator. By using the fixed theorem of cone expansion and compression of norm type in a cone, the existence of positive solutions is established though nonlinear term contains the first derivative of unknown function.     

K 0-group and similarity of operator weighted shifts

By using simultaneous triangularization technique, similarity for operator weighted shifts with finite multiplicity is characterized in terms of K0-group of commutant algebra. The result supports the conjecture posed by Cao et al. in 1999. Moreover, we discuss the relations between similarity and quasi-similarity for operator weighted shifts.