Spectral theory for polynomially demicompact operators

In this article, we introduce the notion of polynomial demicompactness and we use it to give some results on Fredholm operators and to establish a fine description of some essential spectra of a closed densely defined linear operator. Our work is a generalization of many known ones in the literature.

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

Filomat ◽

10.2298/fil1509983a ◽

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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Relatively compact-like perturbations, essential spectra and application

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700010168 ◽

2004 ◽

Vol 77 (1) ◽

pp. 73-90 ◽

Cited By ~ 17

Author(s):

Khalid Latrach ◽

J. Martin Paoli

Keyword(s):

Banach Space ◽

Linear Operator ◽

Neutron Transport ◽

Linear Operators ◽

Essential Spectra ◽

Graph Norm ◽

Bounded Linear ◽

Neutron Transport Equations ◽

Densely Defined ◽

Fredholm Perturbations

AbstractThe purpose of this paper is to provide a detailed treatment of the behaviour of essential spectra of closed densely defined linear operators subjected to additive perturbations not necessarily belonging to any ideal of the algebra of bounded linear operators. IfAdenotes a closed densely defined linear operator on a Banach spaceX, our approach consists principally in considering the class ofA-closable operators which, regarded as operators in ℒ(XA,X) (whereXAdenotes the domain ofAequipped with the graph norm), are contained in the set ofA-Fredholm perturbations (see Definition 1.2). Our results are used to describe the essential spectra of singular neutron transport equations in bounded geometries.

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Formally Normal Operators Having no Normal Extensions

Canadian Journal of Mathematics ◽

10.4153/cjm-1965-098-8 ◽

1965 ◽

Vol 17 ◽

pp. 1030-1040 ◽

Cited By ~ 24

Author(s):

Earl A. Coddington

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Null Space ◽

Normal Operator ◽

Closed Linear Operator ◽

Normal Operators ◽

The Given ◽

Densely Defined ◽

Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

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Solvability conditions for some non-Fredholm operators

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091509000236 ◽

2010 ◽

Vol 54 (1) ◽

pp. 249-271 ◽

Cited By ~ 18

Author(s):

Vitali Vougalter ◽

Vitaly Volpert

Keyword(s):

Spectral Theory ◽

Elliptic Equations ◽

Scattering Theory ◽

Adjoint Equation ◽

Fredholm Operators ◽

Solvability Conditions ◽

Non Fredholm Operators ◽

Schrödinger Type Operators

AbstractWe obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger-type operators. One of the main results of the paper concerns solvability conditions for the equation –Δu+V(x)u–au=fwhere a ≥ 0. The conditions are formulated in terms of orthogonality of the functionfto the solutions of the homogeneous adjoint equation.

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Dirac structures on Hilbert spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171299220972 ◽

1999 ◽

Vol 22 (1) ◽

pp. 97-108 ◽

Cited By ~ 2

Author(s):

A. Parsian ◽

A. Shafei Deh Abad

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Hilbert Spaces ◽

Smooth Manifold ◽

Real Hilbert Space ◽

Dirac Structure ◽

Dirac Structures ◽

Basic Properties ◽

Hilbert Subspace ◽

Densely Defined

For a real Hilbert space(H,〈,〉), a subspaceL⊂H⊕His said to be a Dirac structure onHif it is maximally isotropic with respect to the pairing〈(x,y),(x′,y′)〉+=(1/2)(〈x,y′〉+〈x′,y〉). By investigating some basic properties of these structures, it is shown that Dirac structures onHare in one-to-one correspondence with isometries onH, and, any two Dirac structures are isometric. It is, also, proved that any Dirac structure on a smooth manifold in the sense of [1] yields a Dirac structure on some Hilbert space. The graph of any densely defined skew symmetric linear operator on a Hilbert space is, also, shown to be a Dirac structure. For a Dirac structureLonH, everyz∈His uniquely decomposed asz=p1(l)+p2(l)for somel∈L, wherep1andp2are projections. Whenp1(L)is closed, for any Hilbert subspaceW⊂H, an induced Dirac structure onWis introduced. The latter concept has also been generalized.

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A characterization of some subsets of S-essential spectra of a multivalued linear operator

Colloquium Mathematicum ◽

10.4064/cm135-2-2 ◽

2014 ◽

Vol 135 (2) ◽

pp. 171-186 ◽

Cited By ~ 7

Author(s):

Teresa Álvarez ◽

Aymen Ammar ◽

Aref Jeribi

Keyword(s):

Linear Operator ◽

Essential Spectra ◽

Multivalued Linear Operator

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Derivations, local derivations and atomic boolean subspace lattices

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700040314 ◽

2002 ◽

Vol 66 (3) ◽

pp. 477-486 ◽

Cited By ~ 6

Author(s):

Pengtong Li ◽

Jipu Ma

Keyword(s):

Banach Space ◽

Linear Operator ◽

Closed Linear Operator ◽

Subspace Lattice ◽

Local Derivation ◽

Densely Defined ◽

Subspace Lattices

Let ℒ be an atomic Boolean subspace lattice on a Banach space X. In this paper, we prove that if ℳ is an ideal of Alg ℒ then every derivation δ from Alg ℒ into ℳ is necessarily quasi-spatial, that is, there exists a densely defined closed linear operator T: 𝒟(T) ⊆ X → X with its domain 𝒟(T) invariant under every element of Alg ℒ, such that δ(A) x = (TA – AT) x for every A ∈ Alg ℒ and every x ∈ 𝒟(T). Also, if ℳ ⊆ ℬ(X) is an Alg ℒ-module then it is shown that every local derivation from Alg ℒ into ℳ is necessary a derivation. In particular, every local derivation from Alg ℒ into ℬ(X) is a derivation and every local derivation from Alg ℒ into itself is a quasi-spatial derivation.

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Some special characterisations of Fredholm operators in Banach space

BIBECHANA ◽

10.3126/bibechana.v11i0.10399 ◽

2014 ◽

Vol 11 ◽

pp. 169-174

Author(s):

Mahendra Shahi

Keyword(s):

Banach Space ◽

Linear Operator ◽

Essential Spectrum ◽

Fredholm Operator ◽

Bounded Linear Operator ◽

Global Analysis ◽

Spectral Theorem ◽

Bounded Operators ◽

Fredholm Operators ◽

Bounded Linear

A bounded linear operator which has a finite index and which is defined on a Banach space is often referred to in the literature as a Fredholm operator. Fredholm operators are important for a variety of reasons, one being the role that their index plays in global analysis. The aim of this paper is to prove the spectral theorem for compact operators in refined form and to describe some properties of the essential spectrum of general bounded operators by the use of the theorem of Fredholm operators. For this, we have analysed the Fredholm operator which is defined in a Banach space for some special characterisations. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10399 BIBECHANA 11(1) (2014) 169-174

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Existence and bifurcation of solutions for Fredholm operators with nonlinear perturbations

Nagoya Mathematical Journal ◽

10.1017/s0027763000019887 ◽

1982 ◽

Vol 86 ◽

pp. 249-271 ◽

Cited By ~ 1

Author(s):

Yasuo Niikura

Keyword(s):

Banach Space ◽

Linear Operator ◽

Nonlinear Operator ◽

Real Banach Space ◽

Eigenvalue Problems ◽

Nonlinear Perturbations ◽

Fredholm Operators ◽

Nonlinear Eigenvalue Problems ◽

Bifurcation Of Solutions ◽

Nonlinear Eigenvalue

In this paper we shall discuss nonlinear eigenvalue problems for the equations of the formwhere L is a linear operator on a real Banach space X with non-zero kernel, K(-) is a linear or nonlinear operator on X and M(·, ·) is an operator from X X R into X. Equations of the form (1) arise in various fields of physics and engineering.

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Closed linear operators with domain containing their range

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500022331 ◽

1984 ◽

Vol 27 (2) ◽

pp. 229-233 ◽

Cited By ~ 7

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).

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