Finite sections of band-dominated operators / Steffen Roch.

Format:

Book

Author/Creator:

Roch, Steffen, 1958- author.

Series:

Memoirs of the American Mathematical Society ; no. 895.

Memoirs of the American Mathematical Society, 0065-9266 ; number 895

Language:

English

Subjects (All):

Operator algebras.

Operator theory.

Physical Description:

1 online resource (104 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

The goal of this text is to review recent advances and to present new results in the numerical analysis of the finite sections method for general band and band-dominated operators. The main topics are the stability of the finite sections method and the asymptotic behavior of singular values. The latter topic is closely related with compactness and Fredholm properties of approximation sequences, and the paper can also serve as an introduction into this remarkable field of numerical analysis. Further the author discusses the behavior of approximation numbers, determinants, essential spectra and essential pseudospectra as well as the localization of pseudomodes of finite sections of band-dominated operators.

Contents:

""Contents""; ""Chapter 1. Introduction: Band-dominated operators""; ""Chapter 2. Stability""; ""2.1. Fredholm band-dominated operators and their indices""; ""2.2. Algebras of matrix sequences""; ""2.3. Stability vs. Fredholmness""; ""2.4. Stability of the finite sections method""; ""2.5. Band-dominated operators with slowly oscillating coefficients""; ""2.6. The role of the index""; ""2.7. The C*-algebra of the finite sections method""; ""2.8. Another perspective on localization""; ""Chapter 3. Stable regularizability""; ""3.1. Moore-Penrose invertibility in C*-algebras""

""3.2. Stable regularizability vs. Moore- Penrose invertibility""""3.3. Stable regularizability of the finite sections method""; ""Chapter 4. Compactness""; ""4.1. Compact sequences""; ""4.2. Characterization via singular values""; ""4.3. Central rank characterizations""; ""4.4. Minimal and maximal characterizations""; ""4.5. Compact sequences in S(N) and S(Z)""; ""Chapter 5. Fredholmness""; ""5.1. Fredholm sequences""; ""5.2. Fredholmness and stability""; ""5.3. Fredholmness and stable regularizability""; ""5.4. Fredholmness of the finite sections method""

""5.5. Slowly oscillating coefficients""""Chapter 6. Essential fractality""; ""6.1. Fractality of quotient maps""; ""6.2. J-fractal algebras""; ""6.3. Essential fractality and singular values""; ""6.4. Essential fractality of the finite sections algebras""; ""Chapter 7. Applications""; ""7.1. Approximation numbers""; ""7.2. Rank-preserving discretizations""; ""7.3. Arveson dichotomy: band- dominated operators""; ""7.4. Arveson dichotomy: the general setting""; ""7.5. Essential spectra""; ""7.6. Essential pseudospectra""; ""7.7. Pseudomodes""; ""7.8. Determinants""; ""Bibliography""; ""Index""

Notes:

"Volume 191, number 895 (end of volume)."

Includes bibliographical references (pages 83-85) and index.

Description based on print version record.

ISBN:

1-4704-0501-6