Quasi-banded operators, convolutions with almost periodic or quasi-continuous data, and their approximations

Mascarenhas, Helena; Santos, Pedro A.; Seidel, M.

doi:10.1016/j.jmaa.2014.03.079

Cited by 10 publications

(24 citation statements)

References 14 publications

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“…We want to mention that versions of both results, Theorem 6.6 and Corollary 6.7, are already contained in the literature: In the Hilbert space case they follow directly from (6.6) by a C *algebra argument (as in footnote 5). [21] gives such results while even exceeding the setting of band-dominated operators considerably. The general case X = l p (Z N , X) is studied in [37] and in Section 3.2 of [34].…”

mentioning

confidence: 90%

Essential pseudospectra and essential norms of band-dominated operators

Hagger

Lindner

Seidel

2016

Journal of Mathematical Analysis and Applications

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An operator A on an l p -space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of A in the Calkin algebra determines, for example, the Fredholmness of A, the Fredholm index, the essential spectrum, the essential norm and the so-called essential pseudospectrum of A. This coset can be identified with the collection of all so-called limit operators of A. It is known that this identification preserves invertibility (hence spectra). We now show that it also preserves norms and in particular resolvent norms (hence pseudospectra). In fact we work with a generalization of the ideal of compact operators, so-called P-compact operators, allowing for a more flexible framework that naturally extends to l p -spaces with p ∈ {1, ∞} and/or vector-valued l p -spaces.

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confidence: 90%

Essential pseudospectra and essential norms of band-dominated operators

Hagger

Lindner

Seidel

2016

Journal of Mathematical Analysis and Applications

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“…Proof The

sans-serifW

‐snapshots for all functions under consideration are almost obvious or have already been discussed in . Since

a \in L_{0}^{\infty}

yields

a I \in {scriptK}^{c} = K false(X, scriptP false)

, hence

{a I} \in {scriptJ}^{boldT}

, we see that all

sans-serifH

‐snapshots are zero by Lemma .…”

Section: Setting Up the Algebraic Frameworkmentioning

confidence: 69%

“…In the next section, we return to our concrete setting

X = L^{p} false(double-struckR false)

with the sequence of the (non‐compact) canonical projections

P_{n}

. Note that this approach already served as a key instrument in for the treatment of the finite sections of arbitrary convolution and convolution type operators on the spaces

L^{1} false(double-struckR false)

and

L^{\infty} false(double-struckR false)

, where the situation with the classical approach is even worse: the

P_{n}

do not converge

*

‐strongly there.…”

Section: Some Basic Conceptsmentioning

confidence: 99%

“…Almost obviously, band operators are band‐dominated, and band‐dominated ones are quasi‐banded. Moreover, one can show that both, the set of band‐dominated operators and the set of quasi‐banded operators are closed and inverse closed subalgebras of

L (X, P)

which contain

K (X, P)

as a closed ideal. If

A

from one of these algebras is

scriptP

‐Fredholm, then its

scriptP

‐regularizers are in the same algebra again.…”

Section: Some Basic Conceptsmentioning

confidence: 99%

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Approximation sequences to operators on Banach spaces: a rich approach

Mascarenhas

Santos

Seidel

2017

Journal of London Math Soc

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Criteria for the stability of finite sections of a large class of convolution type operators on L p (R) are obtained. In this class almost all classical symbols are permitted, namely operators of multiplication with functions in [PC, SO, L ∞ 0 ] and convolution operators (as well as Wiener-Hopf and Hankel operators) with symbols in [PC, SO, AP, BUC] p . We use a simpler and more powerful algebraic technique than all previous works: the application of P-theory together with the rich sequences concept and localization. Beyond stability we study Fredholm theory in sequence algebras. In particular, formulas for the asymptotic behavior of approximation numbers and Fredholm indices are given.

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“…Thus, we could also consider the (more general) set of operators which are only subject to this condition, and we still have Theorem 4.3. Such operators are called quasi-banded operators, were introduced in [22] and studied in [21, Section 4], [13]. They form a Banach subalgebra of L(l p (Z, X), P) include all band-dominated operators, but also contain e.g.…”

Section: On Semi-fredholm Band-dominated Operatorsmentioning

confidence: 99%