scholarly journals On the removal of weak compactness arguments in proof mining

2019 ◽  
Vol 354 ◽  
pp. 106728 ◽  
Author(s):  
Fernando Ferreira ◽  
Laurenţiu Leuştean ◽  
Pedro Pinto
Keyword(s):  
Weak Compactness ◽  
Proof Mining ◽  
Compactness Arguments

scholarly journals Combined Mean-Field and Semiclassical Limits of Large Fermionic Systems

2021 ◽  
Vol 182 (2) ◽  
Author(s):  
Li Chen ◽  
Jinyeop Lee ◽  
Matthew Liew
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Mean Field ◽  
Weak Limit ◽  
Weak Compactness ◽  
Three Dimensions ◽  
Uniform Estimates ◽  
Fermionic Systems ◽  
Semiclassical Limits ◽  
Semiclassical Regime

AbstractWe study the time dependent Schrödinger equation for large spinless fermions with the semiclassical scale $$\hbar = N^{-1/3}$$ ħ = N - 1 / 3 in three dimensions. By using the Husimi measure defined by coherent states, we rewrite the Schrödinger equation into a BBGKY type of hierarchy for the k particle Husimi measure. Further estimates are derived to obtain the weak compactness of the Husimi measure, and in addition uniform estimates for the remainder terms in the hierarchy are derived in order to show that in the semiclassical regime the weak limit of the Husimi measure is exactly the solution of the Vlasov equation.

scholarly journals The spectral theorem for well-bounded operators

1993 ◽  
Vol 54 (3) ◽  
pp. 334-351 ◽  
Author(s):  
Ian Doust ◽  
Qiu Bozhou
Keyword(s):  
Functional Calculus ◽  
Continuous Functions ◽  
Weak Compactness ◽  
Compact Interval ◽  
Spectral Theorem ◽  
Type B ◽  
Bounded Operators ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous

AbstractWell-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

Weak compactness of Fréchet-derivatives: application to composition operators

1980 ◽  
Vol 29 (4) ◽  
pp. 399-406
Author(s):  
Peter Dierolf ◽  
Jürgen Voigt
Keyword(s):  
Banach Spaces ◽  
Integral Equations ◽  
Sobolev Spaces ◽  
Composition Operators ◽  
Weak Compactness ◽  
Nonlinear Integral Equations ◽  
Weakly Compact ◽  
Frechet Derivatives ◽  
Fréchet Derivatives ◽  
Compact Map

AbstractWe prove a result on compactness properties of Fréchet-derivatives which implies that the Fréchet-derivative of a weakly compact map between Banach spaces is weakly compact. This result is applied to characterize certain weakly compact composition operators on Sobolev spaces which have application in the theory of nonlinear integral equations and in the calculus of variations.

Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions

1999 ◽  
Vol 42 (2) ◽  
pp. 139-148 ◽  
Author(s):  
José Bonet ◽  
Paweł Dománski ◽  
Mikael Lindström
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Composition Operators ◽  
Essential Norm ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Point Evaluation ◽  
Weighted Banach Spaces ◽  
Spaces Of Analytic Functions

AbstractEvery weakly compact composition operator between weighted Banach spaces of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces.

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