The Spectral Resolution of Some Non-Selfadjoint Partial Differential Operators

1975 ◽  
Vol 27 (6) ◽  
pp. 1316-1322
Author(s):  
Gustavus E. Huige
Keyword(s):  
Spectral Resolution ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Linear Operators ◽  
Partial Differential ◽  
Partial Differential Operators ◽  
Densely Defined

Let ffl and be Hilbert spaces, the set of all densely defined linear operators from to , and its subset of bounded ones. Let and σ(T) denote the adjoint, range, domain, closure and spectrum of T respectively. Ri(z) will denote the resolvent (z — Ti)-1.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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