Convolution Operators and Factorization of Almost Periodic Matrix Functions

Böttcher, Albrecht; Karlovich, Yuri I.; Спитковский, И. М.

doi:10.1007/978-3-0348-8152-4

Cited by 133 publications

(218 citation statements)

References 0 publications

Supporting

Mentioning

215

Contrasting

Unclassified

Order By: Relevance

“…Further, if (α − µ)/(µ + σ) is rational, then the distances between the points of sp(g) are commensurable. This again guarantees the AP P factorability, with factorization formulas given in [15] and [2,Section 14.4]. We will therefore suppose that…”

Section: Almost Periodic Symbols and Factorizationmentioning

confidence: 94%

“…For matrix functions G ∈ (AP W ) n×n this sufficient invertibility condition is also necessary [2,Theorem 5.16]. Moreover, (2.2) is then automatically an AP W factorization of G. This is the main reason because of which the AP factorization of matrices (1.2) with g ∈ AP W is of interest.…”

Section: Almost Periodic Symbols and Factorizationmentioning

confidence: 99%

“…If α or σ are non-positive, then sp(g) lies to one side of the origin. Either way, G is then AP P factorable, and an explicit factorization was constructed in [12], see also [2], Sections 14.1 and 14.3. Further, if (α − µ)/(µ + σ) is rational, then the distances between the points of sp(g) are commensurable.…”

Section: Almost Periodic Symbols and Factorizationmentioning

confidence: 99%

“…[2]), focusing mainly on the case where the non-diagonal function g is an almost periodic polynomial, i.e., g ∈ AP P . Recall that AP P consists, by definition, of all finite linear combinations…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Toeplitz operators of finite interval type and the table method

Câmara

Diogo

Spitkovsky

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

We solve a Riemann-Hilbert problem with almost periodic coefficient G, associated to a Toeplitz operator T G in a class which is closely connected to finite interval convolution equations, based on a generalization of the socalled table method. The explicit determination of solutions to that problem allows one to establish necessary and sufficient conditions for the invertibility of the corresponding Toeplitz operator, and to determine an appropriate factorization of G, providing explicit formulas for the inverse of T G . Some unexpected properties of the Fourier spectrum of the solutions are revealed which are not apparent through other approaches to the same problem.

show abstract

Section: Almost Periodic Symbols and Factorizationmentioning

confidence: 94%

Section: Almost Periodic Symbols and Factorizationmentioning

confidence: 99%

Section: Almost Periodic Symbols and Factorizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Toeplitz operators of finite interval type and the table method

Câmara

Diogo

Spitkovsky

2015

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…Applying the inverse closedness of the algebra SAP p in L ∞ (R) (see [2,Proposition 19.4]), we immediately get the following. …”

Section: Convolution Operatorsmentioning

confidence: 99%

Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Karlovich

Lebre

2011

Integr. Equ. Oper. Theory

Self Cite

View full text Add to dashboard Cite

Abstract. Suppose α is an orientation-preserving diffeomorphism (shift) of R + = (0, ∞) onto itself with the only fixed points 0 and ∞. In [6] we found sufficient conditions for the Fredholmness of the singular integral operator with shiftacting on L p (R + ) with 1 < p < ∞, where P ± = (I ± S)/2, S is the Cauchy singular integral operator, and W α f = f • α is the shift operator, under the assumptions that the coefficients a, b, c, d and the derivative α ′ of the shift are bounded and continuous on R + and may admit discontinuities of slowly oscillating type at 0 and ∞. Now we prove that those conditions are also necessary.

show abstract

Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data

Bastos¹,

Bravo²,

Karlovich

2004

Mathematische Nachrichten

View full text Add to dashboard Cite

A symbol calculus for the smallest Banach subalgebra A [SO,P C] n×n means the C * -algebra generated by all slowly oscillating (SO) and all piecewise continuous (P C) n × n matrix functions, and [SOp, P Cp] n×n is a Fourier multiplier analogue of [SO, P C] n×n on Lp(R). As a result, a Fredholm criterion for the operators A ∈ A [SO,P C] is established. The study is based on the compactness of the commutators AW a,b − W a,b A where A ∈ A [SO,P C] , a ∈ SO, and b ∈ SOp, on the Allan-Douglas local principle, and on the two projections theorem.

show abstract

Convolution Operators and Factorization of Almost Periodic Matrix Functions

Abstract: specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

Cited by 133 publications

References 0 publications

Toeplitz operators of finite interval type and the table method

Toeplitz operators of finite interval type and the table method

Necessary Conditions for Fredholmness of Singular Integral Operators with Shifts and Slowly Oscillating Data

Symbol calculus and Fredholmness for a Banach algebra of convolution type operators with slowly oscillating and piecewise continuous data

Contact Info

Product

Resources

About