LOCAL SMITH FORM AND EQUIVALENCE FOR ONE-PARAMETER FAMILIES OF FREDHOLM OPERATORS OF INDEX ZERO

2005 ◽  
Author(s):  
JULIÁN LÓPEZ-GÓMEZ ◽  
CARLOS MORA-CORRAL
Keyword(s):  
Fredholm Operators ◽  
Index Zero

Global Perturbation of Nonlinear Eigenvalues

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Perturbation Theory ◽  
Spectral Parameter ◽  
Classical Theory ◽  
General Setting ◽  
Perturbation Parameter ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Index Zero ◽  
Finite Dimensional ◽  
Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

Uniqueness of the algebraic multiplicity

2004 ◽  
Vol 134 (5) ◽  
pp. 985-990
Author(s):  
Carlos Mora-Corral
Keyword(s):  
Banach Space ◽  
Complex Variable ◽  
Smooth Function ◽  
Algebraic Multiplicity ◽  
Fredholm Operators ◽  
Index Zero ◽  
The Family

Given a smooth function 𝔏 of a real or complex variable and taking its values in the class of Fredholm operators of index zero in a Banach space, there are some available definitions in the literature of algebraic multiplicity of the family 𝔏 at a point x0 of the parameter at which the operator 𝔏(x0) becomes non-invertible. The purpose of this paper is to suggest an axiomatic for the multiplicity and to prove that the algebraic multiplicity is uniquely determined by a few of its properties, independently of its construction.

scholarly journals Axiomatization of the degree of Fitzpatrick, Pejsachowicz and Rabier

2021 ◽  
Vol 24 (1) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Uniqueness Theorems ◽  
Algebraic Multiplicity ◽  
Fredholm Operators ◽  
Index Zero ◽  
Crucial Step ◽  
Fredholm Maps

AbstractIn this paper, we prove an analogue of the uniqueness theorems of Führer [15] and Amann and Weiss [1] to cover the degree of Fredholm operators of index zero constructed by Fitzpatrick, Pejsachowicz and Rabier [13], whose range of applicability is substantially wider than for the most classical degrees of Brouwer [5] and Leray–Schauder [22]. A crucial step towards the axiomatization of this degree is provided by the generalized algebraic multiplicity of Esquinas and López-Gómez [8, 9, 25], $$\chi $$ χ , and the axiomatization theorem of Mora-Corral [28, 32]. The latest result facilitates the axiomatization of the parity of Fitzpatrick and Pejsachowicz [12], $$\sigma (\cdot ,[a,b])$$ σ ( · , [ a , b ] ) , which provides the key step for establishing the uniqueness of the degree for Fredholm maps.

Economical finite rank perturbations of semi-Fredholm operators

1988 ◽  
Vol 198 (3) ◽  
pp. 431-434 ◽  
Author(s):  
M�che�l � Searc�id
Keyword(s):  
Finite Rank ◽  
Fredholm Operators

scholarly journals Stability of essential approximate point spectrum and essential defect spectrum of linear operator

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1983-1994
Author(s):  
Aymen Ammar ◽  
Mohammed Dhahri ◽  
Aref Jeribi
Keyword(s):  
Linear Operator ◽  
Measure Of Noncompactness ◽  
Point Spectrum ◽  
Fredholm Operators ◽  
Approximate Point Spectrum ◽  
Densely Defined

In the present paper, we use the notion of measure of noncompactness to give some results on Fredholm operators and we establish a fine description of the essential approximate point spectrum and the essential defect spectrum of a closed densely defined linear operator.

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