Compactness, Fredholm Alternative and Spectrum I Compactness Definition We say that E⊆ Xis compact if from eavery open covering of Eit is possible to extract a finite subcovering of E Definition A subset E of a normed space X is sequentially compact if for every sequence {xk}⊂Ethere exists a subsequence {xk}, converging in X Theorem 1
Here, we prove the basic Fredholm alternative on Banach spaces, that for compact T and non-zero 2C, either T is a bijection, or has closed image of codimension equal to the dimension of its kernel In particular, the only non-zero spectrum is point spectrum [2]
[1 11] Discreteness of spectrum of compact operators [1 12] Claim: For T a compact operator on a Hilbert the non-zero spectrum (if any) is point spectrum The number of eigenvalues outside a given disk j j ris nite for r>0, and always 0 is in the spectrum
35 Compact and Fredholm Operators and the Spectral Theorem In this section Hand Bwill be Hilbert spaces Typically Hand Bwill be separable, but we will not assume this until it is needed later 35 1 Compact Operators Proposition 35 1 Let Mbe a finite dimensional subspace of a Hilbert space H then (1) Mis complete (hence closed)
4 8 Compactness of the resolvent 105 4 9 The Fredholm alternative 106 4 10 The spectrum of a self-adjoint elliptic operator 108 4 11 Interior regularity 110 4 12 Boundary regularity 114 4 13 Some further perspectives 116 Appendix 119 4 A Heat flow 119 4 B Operators on Hilbert spaces 121 4 C Difference quotients 124 Chapter 5
4 8 Compactness of the resolvent 101 4 9 The Fredholm alternative 102 4 10 The spectrum of a self-adjoint elliptic operator 104 4 11 Interior regularity 105 4 12 Boundary regularity 109 4 13 Some further perspectives 112 4 A Heat ow 114 4 B Operators on Hilbert spaces 116 4 C Di erence quotients and weak derivatives 119 Chapter 5 The
The transpose of a compact operator is compact—The Fredholm alternative Historical note, 244 22 Examples of Compact Operators 245 22 1 Compactness criteria, 245 Arzela-Ascoli compactness criterion—Rellich compactness criterion 22 2 Integral operators, 246 Hilbert-Schmidt operators 22 3 The inverse of elliptic partial differential operators
Lecture 28 The Spectrum of a Linear Map Lecture 29 Some examples Part 8 Compact Linear Maps Lecture 30 Compact Maps Lecture 31 Fredholm alternative Lecture 32 Spectral Theory of Compact Maps Homework III Part 9 Compact Linear Maps in Hilbert Space Lecture 33 Compact Symmetric Operators Lecture 34 Min-Max Lecture 35 Functional calculus
Compactness of the Identity Operator p 271 Properties p 272 Approximation by Finite-Rank Operators p 275 Fredholm Alternative p 278 Spectrum of a Compact Operator, Riesz-Schauder Theorem p 279 Application: Fredholm Integral Equations of the Second Kind p 283 Problems p 284 Elements of Spectral Theory in a Hilbert Space Setting p 293
Research Article Some Compactness and Interpolation Results for Linear Boltzmann Equation NadjehRedjel 1,2 andAbdelkaderDehici 1,2 Laboratory of Informatics and Mathematics, University of Souk
The image ofthe closed unit ball under a compact linear transformation is The spectrum of a compact normal operator is cOllntable; Fredholm alternative
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We will obtain results on the nature of the spectrum of a compact operator after proving some obtain the classical Fredholm alternative for compact operators
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