scholarly journals UNBOUNDED B-FREDHOLM OPERATORS ON HILBERT SPACES

2008 ◽  
Vol 51 (2) ◽  
pp. 285-296 ◽  
Author(s):  
M. Berkani ◽  
N. Castro-González
Keyword(s):  
Fredholm Operator ◽  
Hilbert Spaces ◽  
Closed Operator ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Spectral Mapping ◽  
Nilpotent Operator ◽  
Densely Defined ◽  
Closed Linear Operators

AbstractThis paper is concerned with the study of a class of closed linear operators densely defined on a Hilbert space $H$ and called B-Fredholm operators. We characterize a B-Fredholm operator as the direct sum of a Fredholm closed operator and a bounded nilpotent operator. The notion of an index of a B-Fredholm operator is introduced and a characterization of B-Fredholm operators with index $0$ is given in terms of the sum of a Drazin closed operator and a finite-rank operator. We analyse the properties of the powers $T^m$ of a closed B-Fredholm operator and we establish a spectral mapping theorem.

scholarly journals Sets of periods for chaotic linear operators

2021 ◽  
Vol 115 (2) ◽  
Author(s):  
J. A. Conejero ◽  
F. Martínez-Giménez ◽  
A. Peris ◽  
F. Rodenas
Keyword(s):  
Banach Spaces ◽  
Hilbert Spaces ◽  
Complete Characterization ◽  
Linear Operators

AbstractWe provide a complete characterization of the possible sets of periods for Devaney chaotic linear operators on Hilbert spaces. As a consequence, we also derive this characterization for linearizable maps on Banach spaces.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

scholarly journals Weighted composition operators on functional Hilbert spaces

1985 ◽  
Vol 31 (1) ◽  
pp. 117-126 ◽  
Author(s):  
R.K. Singh ◽  
R. David Chandra Kumar
Keyword(s):  
Hilbert Spaces ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Linear Operators ◽  
Weighted Composition Operators ◽  
Bounded Linear ◽  
Atomic Measure ◽  
Weighted Composition ◽  
Functional Hilbert Spaces

Let X be a non-empty set and let H(X) denote a Hibert space of complex-valued functions on X. Let T be a mapping from X to X and θ a mapping from X to C such that for all f in H(X), f ° T is in H(x) and the mappings CT taking f to f ° T and M taking f to θ.f are bounded linear operators on H(X). Then the operator CTMθ is called a weighted composition operator on H(X). This note is a report on the characterization of weighted composition operators on functional Hilbert spaces and the computation of the adjoint of such operators on L2 of an atomic measure space. Also the Fredholm criteria are discussed for such classes of operators.

Characterization of Closed Densely Defined Semi-Browder Linear Operators

2012 ◽  
Vol 7 (6) ◽  
pp. 1775-1786 ◽  
Author(s):  
Teresa Alvarez ◽  
Fatma Fakhfakh ◽  
Maher Mnif
Keyword(s):  
Linear Operators ◽  
Densely Defined

A characterization of the essential approximation pseudospectrum on a Banach space

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3599-3610 ◽  
Author(s):  
Aymen Ammar ◽  
Aref Jeribi ◽  
Kamel Mahfoudhi
Keyword(s):  
Banach Space ◽  
Approximation Theory ◽  
Linear Operators ◽  
The Stability ◽  
Densely Defined

One impetus for writing this paper is the issue of approximation pseudospectrum introduced by M. P. H.Wolff in the journal of approximation theory (2001). The latter study motivates us to investigate the essential approximation pseudospectrum of closed, densely defined linear operators on a Banach space. We begin by defining it and then we focus on the characterization, the stability and some properties of these pseudospectra.

Well-posedness of Third Order Differential Equations in Hölder Continuous Function Spaces

2018 ◽  
Vol 62 (4) ◽  
pp. 715-726
Author(s):  
Shangquan Bu ◽  
Gang Cai
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Function Spaces ◽  
Linear Operators ◽  
Well Posedness ◽  
Third Order ◽  
Hölder Continuous ◽  
Vector Valued ◽  
Closed Linear Operators

AbstractIn this paper, by using operator-valued ${\dot{C}}^{\unicode[STIX]{x1D6FC}}$-Fourier multiplier results on vector-valued Hölder continuous function spaces, we give a characterization of the $C^{\unicode[STIX]{x1D6FC}}$-well-posedness for the third order differential equations $au^{\prime \prime \prime }(t)+u^{\prime \prime }(t)=Au(t)+Bu^{\prime }(t)+f(t)$, ($t\in \mathbb{R}$), where $A,B$ are closed linear operators on a Banach space $X$ such that $D(A)\subset D(B)$, $a\in \mathbb{C}$ and $0<\unicode[STIX]{x1D6FC}<1$.

scholarly journals Closed linear operators with domain containing their range

1984 ◽  
Vol 27 (2) ◽  
pp. 229-233 ◽  
Author(s):  
Schôichi Ôta
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Linear Operators ◽  
Closed Linear Operator ◽  
Unbounded Operators ◽  
Densely Defined ◽  
Closed Linear Operators

In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).

On the null spaces of linear Fredholm operators depending on several parameters

1978 ◽  
Vol 84 (1) ◽  
pp. 131-142 ◽  
Author(s):  
M. Shearer
Keyword(s):  
Banach Spaces ◽  
Fredholm Operator ◽  
Bifurcation Theory ◽  
Null Space ◽  
Parameter Family ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

AbstractLet X, Y be real Banach spaces, and let = {f(λ):λ ∈ m} be an m-parameter family of bounded linear operators from X to Y, with f(λ) depending continuously on λ. The cases m = 1 and m = 2 are considered, and conditions on are found which determine the null space of f(λ) for all λ near a given λ0 such that f(λ0): X → Y is a Fredholm operator. The results obtained are shown to be of particular interest in perturbed bifurcation theory.

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