Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite sections

Rabinovich, Vladimir; Roch, Steffen; Silbermann, Bernd

doi:10.1007/bf01299850

Cited by 58 publications

(54 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One can easily check the following properties of limit operators (cf., e.g., [3, Proposition 6.1], [17,Proposition 2.12]). A, B, A n ∈ B(L 2 (Π)), n ∈ N, and τ ∈ {0, ∞}.…”

Section: Canonical Form Of Local Algebrasmentioning

confidence: 99%

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

Karlovich

Pessoa

2005

Integr. equ. oper. theory

View full text Add to dashboard Cite

The C * -algebra A generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L 2 (Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C * -algebras associated with points z ∈ Π ∪ ∂Π and pairs (z, λ) ∈ ∂Π × R. We construct a symbol calculus for unital C * -algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈ ∂Π being the discontinuity points of coefficients. A symbol calculus for the C * -algebra A and a Fredholm criterion for the operators A ∈ A are obtained. Finally, a C * -algebra isomorphism between the quotient algebra A π = A/K, where K is the ideal of compact operators, and its analogue A π D for the unit disk is constructed. Mathematics Subject Classification (2000). Primary 47A53, 47L15; Secondary 47G10, 47L30.

show abstract

“…One can easily check the following properties of limit operators (cf., e.g., [3, Proposition 6.1], [17,Proposition 2.12]). A, B, A n ∈ B(L 2 (Π)), n ∈ N, and τ ∈ {0, ∞}.…”

Section: Canonical Form Of Local Algebrasmentioning

confidence: 99%

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

Karlovich

Pessoa

2005

Integr. equ. oper. theory

View full text Add to dashboard Cite

show abstract

“…We also learned of very recent work of Rabinovich [66], based in part on [47,61,64,65,67,68], using the theory of Fredholm operators to obtain results on essential spectrum as a union of spectra of suitable limits at infinity.…”

Section: Introductionmentioning

confidence: 99%

The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators

Simon

2006

J. Anal. Math.

133

View full text Add to dashboard Cite

Abstract. We provide a very general result that identifies the essential spectrum of broad classes of operators as exactly equal to the closure of the union of the spectra of suitable limits at infinity. Included is a new result on the essential spectra when potentials are asymptotic to isospectral tori. We also recover with a unified framework the HVZ theorem and Krein's results on orthogonal polynomials with finite essential spectra.

show abstract

“…In Section 2, we recall some auxiliary material concerning the Wiener algebra of discrete operators on Z N and the limit operators method. Our presentation will follow the papers [20,21]. A comprehensive account of this material can be found in [22].…”

Section: Introductionmentioning

confidence: 99%

The essential spectrum of Schrödinger operators on lattices

Rabinovich¹,

Roch²

2006

J. Phys. A: Math. Gen.

View full text Add to dashboard Cite

The paper is devoted to the study of the essential spectrum of discrete Schrödinger operators on the lattice Z N by means of the limit operators method. This method has been applied by one of the authors to describe the essential spectrum of (continuous) electromagnetic Schrödinger operators, square-root Klein-Gordon operators, and Dirac operators under quite weak assumptions on the behavior of the magnetic and electric potential at infinity. The present paper is aimed to illustrate the applicability and efficiency of the limit operators method to discrete problems as well.We consider the following classes of the discrete Schrödinger operators: 1) operators with slowly oscillating at infinity potentials, 2) operators with periodic and semi-periodic potentials; 3) Schrödinger operators which are discrete quantum analogs of the acoustic propagators for waveguides; 4) operators with potentials having an infinite set of discontinuities; and 5) three-particle Schrödinger operators which describe the motion of two particles around a heavy nuclei on the lattice Z 3 .

show abstract

Band-dominated operators with operator-valued coefficients, their Fredholm properties and finite sections

Cited by 58 publications

References 17 publications

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

The Essential Spectrum of Schrödinger, Jacobi, and CMV Operators

The essential spectrum of Schrödinger operators on lattices

Contact Info

Product

Resources

About