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Representation of operators by kernels.

Peter Stollmann (1991)

Collectanea Mathematica

We prove that differences of order-continuous operators acting between function spaces can be represented with a pseudo-kernel, proved the underlying measure spaces satisfy certain (rather weak) conditions. To see that part of these conditions are necessary, we show that the strict localizability of a measure space can be characterized by the existence of a pseudo-kernel for a certain operator.

Representing non-weakly compact operators

Manuel González, Eero Saksman, Hans-Olav Tylli (1995)

Studia Mathematica

For each S ∈ L(E) (with E a Banach space) the operator R(S) ∈ L(E**/E) is defined by R(S)(x** + E) = S**x** + E(x** ∈ E**). We study mapping properties of the correspondence S → R(S), which provides a representation R of the weak Calkin algebra L(E)/W(E) (here W(E) denotes the weakly compact operators on E). Our results display strongly varying behaviour of R. For instance, there are no non-zero compact operators in Im(R) in the case of L 1 and C(0,1), but R(L(E)/W(E)) identifies isometrically with...

Reproducing kernels, Engliš algebras and some applications

Mubariz T. Karaev, Mehmet Gürdal, Mualla Birgül Huban (2016)

Studia Mathematica

We introduce the notion of Engliš algebras, defined in terms of reproducing kernels and Berezin symbols. Such algebras were apparently first investigated by Engliš (1995). Here we give some new results on Engliš C*-algebras on abstract reproducing kernel Hilbert spaces and some applications to various questions of operator theory. In particular, we give applications to Riccati operator equations, zero Toeplitz products, and the existence of invariant subspaces for some operators.

Résolution d'équations aux dérivées partielles dans des espaces de distributions d'ordre de régularité variable

André Unterberger (1971)

Annales de l'institut Fourier

L’objet de cet article est de prouver des théorèmes du genre suivant : “Soient P un opérateur différentiel sur R n , ρ une fonction C ∞ à valeurs réelles, k un nombre réel et u une distribution à support compact : alors, si P u ∈ H ρ , u ∈ H ρ + k ” ; l’espace H ρ est ici l’espace de Sobolev “d’ordre variable” associé à ρ  ; bien entendu, il faut des hypothèses sur P , ρ et k . Les cas traités sont :1) certains opérateurs à coefficients variables déjà considérés dans le chapitre VIII du livre de L. Hörmander ;2) tous les opérateurs...

Resonant delocalization for random Schrödinger operators on tree graphs

Michael Aizenman, Simone Warzel (2013)

Journal of the European Mathematical Society

We analyse the spectral phase diagram of Schrödinger operators T + λ V on regular tree graphs, with T the graph adjacency operator and V a random potential given by i i d random variables. The main result is a criterion for the emergence of absolutely continuous ( a c ) spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials a c spectrum appears at arbitrarily weak disorder ( λ ≪ 1 ) in an energy regime which extends beyond the spectrum of ∼ T . Incorporating...

Retractions onto the Space of Continuous Divergence-free Vector Fields

Philippe Bouafia (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

We prove that there does not exist a uniformly continuous retraction from the space of continuous vector fields onto the subspace of vector fields whose divergence vanishes in the distributional sense. We then generalise this result using the concept of m -charges, introduced by De Pauw, Moonens, and Pfeffer: on any subset X ⊆ ℝ n satisfying a mild geometric condition, there is no uniformly continuous representation operator for m -charges in X .

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