scholarly journals Some Spaces are not the Domain of a Closed Linear Operator in a Banach Space

1980 ◽  
Vol 23 (4) ◽  
pp. 501-503
Author(s):  
Peter Dierolf ◽  
Susanne Dierolf
Keyword(s):  
Banach Space ◽  
Differential Operator ◽  
Linear Operator ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Closed Linear Operator ◽  
Partial Differential ◽  
Minimal Operator ◽  
Constant Coefficients

Let be a linear partial differential operator with C∞- coefficients. The study of P(∂) as an operator in L2(ℝn) usually starts with the investigation of the minimal operator P0 which is the closure of P(∂) acting on . In the case of constant coefficients it is known that the domain D(P0) of P0 at least contains the space (cf. Schechter [4, p. 58, Lemma 1.2]).

scholarly journals Note on Hypoellipticity of a First Order Linear Partial Differential Operator

1968 ◽  
Vol 32 ◽  
pp. 323-330
Author(s):  
Yoshio Kato
Keyword(s):  
Analytic Function ◽  
Differential Operator ◽  
Differentiable Function ◽  
Partial Differential Operator ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
First Order ◽  
Open Set ◽  
Euclidian Space ◽  
Infinitely Differentiable Function

Let Ω be a domain in the (n + 1)-dimensional euclidian space Rn+1. A linear partial differential operator P with coefficients in C∞(Ω) (resp. in Cω(Ω)) will be termed hypoelliptic (resp. analytic-hypoelliptic) in Ω if a distribution u on Ω (i.e. u ∈ D′(Ω)) is an infinitely differentiable function (resp. an analytic function) in every open set of Ω where Pu is an infinitely differentiable function (resp. an analytic function).

Faddeev Equations in the Functional Quantum Theory of the Non-Linear Spinorfield

1980 ◽  
Vol 35 (9) ◽  
pp. 964-972
Author(s):  
U. Ramacher
Keyword(s):  
Quantum Theory ◽  
Differential Operator ◽  
Effective Potential ◽  
Partial Differential Operator ◽  
Functional Space ◽  
Baryon Number ◽  
Linear Partial Differential Operator ◽  
Partial Differential ◽  
Faddeev Equations ◽  
Polarization Cloud

Abstract Starting with a linear partial differential operator for a certain system of transition amplitudes associated to a state |a> with baryon number ρ(a) we derive an equation in the ρ(a)-sector of the Functional space which provides the state |a> with an effective potential caused by the polarization cloud. With regard to the needs of n-body scattering we then undertake a cluster decomposition of the effective potential. In particular, we derive the relativistic analoga of Lippmann-Schwinger-and Faddeev equations.

scholarly journals Surjectivity of linear operators from a Banach space into itself

2007 ◽  
Vol 76 (1) ◽  
pp. 143-154 ◽  
Author(s):  
Dimosthenis Drivaliaris ◽  
Nikos Yannakakis
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Differential Operator ◽  
Partial Differential Operator ◽  
Second Order ◽  
Linear Operators ◽  
Partial Differential ◽  
Space Case

We show that linear operators from a Banach space into itself which satisfy some relaxed strong accretivity conditions are invertible. Moreover, we characterise a particular class of such operators in the Hilbert space case. By doing so we manage to answer a problem posed by B. Ricceri, concerning a linear second order partial differential operator.

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