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A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces

Katrin Schumacher (2009)

Czechoslovak Mathematical Journal

Given a domain Ω of class C k , 1 , k , we construct a chart that maps normals to the boundary of the half space to normals to the boundary of Ω in the sense that ( - x n ) α ( x ' , 0 ) = - N ( x ' ) and that still is of class C k , 1 . As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to k on domains of class C k , 1 . The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.

A commutant lifting theorem on analytic polyhedra

Calin Ambrozie, Jörg Eschmeier (2005)

Banach Center Publications

In this note a commutant lifting theorem for vector-valued functional Hilbert spaces over generalized analytic polyhedra in ℂⁿ is proved. Let T be the compression of the multiplication tuple M z to a *-invariant closed subspace of the underlying functional Hilbert space. Our main result characterizes those operators in the commutant of T which possess a lifting to a multiplier with Schur class symbol. As an application we obtain interpolation results of Nevanlinna-Pick and Carathéodory-Fejér type...

An operator-theoretic approach to truncated moment problems

Raúl Curto (1997)

Banach Center Publications

We survey recent developments in operator theory and moment problems, beginning with the study of quadratic hyponormality for unilateral weighted shifts, its connections with truncated Hamburger, Stieltjes, Hausdorff and Toeplitz moment problems, and the subsequent proof that polynomially hyponormal operators need not be subnormal. We present a general elementary approach to truncated moment problems in one or several real or complex variables, based on matrix positivity and extension. Together...

Analytic formulas for the hyperbolic distance between two contractions

Ion Suciu (1997)

Annales Polonici Mathematici

In this paper we give some analytic formulas for the hyperbolic (Harnack) distance between two contractions which permit concrete computations in several situations, including the finite-dimensional case. The main consequence of these formulas is the proof of the Schwarz-Pick Lemma. It modifies those given in [13] by the avoidance of a general Schur type formula for contractive analytic functions, more exactly by reducing the case to the more manageable situation when the function takes as values...

Calcul fonctionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux)

Pascal Auscher, Philippe Tchamitchian (1995)

Annales de l'institut Fourier

Dans cet article, on considère les opérateurs différentiels T = b ( x ) D ( a ( x ) D ) , où a ( x ) et b ( x ) sont deux fonctions mesurables, bornées et accrétives, et D = - i d d x . Les résultats principaux portent sur les propriétés fonctionnelles de T , de sa racine carrée, avec applications à l’équation elliptique t 2 u - T u = 0 sur × [ 0 , + [ . On démontre que T 1 / 2 D - 1 est un opérateur de Calderón-Zygmund qui dépend analytiquement du couple ( a , b ) . Les estimations ponctuelles optimales sur le noyau du semi-groupe exp ( - t L 1 / 2 ) et le calcul fonctionnel permettent de développer une théorie...

Common extensions for linear operators

Rodica-Mihaela Dăneţ (2011)

Banach Center Publications

The main meaning of the common extension for two linear operators is the following: given two vector subspaces G₁ and G₂ in a vector space (respectively an ordered vector space) E, a Dedekind complete ordered vector space F and two (positive) linear operators T₁: G₁ → F, T₂: G₂ → F, when does a (positive) linear common extension L of T₁, T₂ exist? First, L will be defined on span(G₁ ∪ G₂). In other results, formulated in the line of the Hahn-Banach extension theorem, the common...

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