Algebras Of Singular Integral Operators Research Articles

The spectrum of the $C^*$-algebra of singular integral operators with semi-almost periodic coefficients

Изучается $C^*$-алгебра, порожденная одномерными сингулярными интегральными операторами с полу-почти-периодическими коэффициентами. Описан примитивный спектр этой $C^*$-алгебры, т.е. перечислены все ее примитивные идеалы и описана топология Джекобсона. Библиография: 27 названий.

On C*-algebras of singular integral operators with PQC coefficients and shifts with fixed points

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the -algebra generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator and by the unitary weighted shift operators , acting on the space over the unit circle . Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms with finite sets of discontinuities for their derivatives , which acts topologically freely on , where is the interior of the nonempty closed set composed by all common fixed points for all shifts , with boundary of zero Lebesgue measure. A Fredholm symbol calculus for the -algebra is constructed and a Fredholm criterion for the operators is established.

Algebras of Singular Integral Operators with Kernels Controlled by Multiple Norms

The purpose of this paper is to study algebras of singular integral operators on $\mathbb{R}^{n}$ and nilpotent Lie groups that arise when one considers the composition of Calder\'on-Zygmund operators with different homogeneities, such as operators that occur in sub-elliptic problems and those arising in elliptic problems. For example, one would like to describe the algebras containing the operators related to the Kohn-Laplacian for appropriate domains, or those related to inverses of H\ormander sub-Laplacians, when these are composed with the more standard class of pseudo-differential operators. The algebras we study can be characterized in a number of different but equivalent ways, and consist of operators that are pseudo-local and bounded on $L^{p}$ for $1<p<\infty$. While the usual class of Calder\'on-Zygmund operators is invariant under a one-parameter family of dilations, the operators we study fall outside this class, and reflect a multi-parameter structure.

A [formula omitted]-algebra of singular integral operators with shifts admitting distinct fixed points

Representations on Hilbert spaces for a nonlocal C⁎-algebra B of singular integral operators with piecewise slowly oscillating coefficients extended by a group of unitary shift operators are constructed. The group of unitary shift operators Ug in the C⁎-algebra B is associated with a discrete amenable group G of orientation-preserving piecewise smooth homeomorphisms g:T→T that acts topologically freely on T and admits distinct fixed points for different shifts. A C⁎-algebra isomorphism of the quotient C⁎-algebra B/K, where K is the ideal of compact operators, onto a C⁎-algebra of Fredholm symbols is constructed by applying the local-trajectory method, spectral measures and a lifting theorem. As a result, a Fredholm symbol calculus for the C⁎-algebra B or, equivalently, a faithful representation of the quotient C⁎-algebra B/K on a suitable Hilbert space is constructed and a Fredholm criterion for the operators B∈B is established.

Wiener’s lemma for singular integral operators of Bessel potential type

In this paper, we introduce an algebra of singular integral operators containing Bessel potentials of positive order, and show that the corresponding unital Banach algebra is an inverse-closed Banach subalgebra of \({\mathcal {B}} (L^q_w)\), the Banach algebra of all bounded operators on the weighted space \(L_w^q\), for all \(1\le q<\infty \) and Muckenhoupt \(A_q\)-weights \(w\).

A Nonlocal C*-Algebra of Singular Integral Operators with Shifts Having Periodic Points

Representations on Hilbert spaces for a nonlocal C*-algebra \({{\mathfrak {B}}}\) of singular integral operators with piecewise slowly oscillating coefficients and unitary shift operators are constructed. The group of unitary shift operators Ug of the C*-algebra \({{\mathfrak {B}}}\) is associated with an amenable discrete group of homeomorphisms \({g:{\mathbb{T}}\to{\mathbb{T}}}\) that have piecewise continuous derivatives and the same nonempty set of periodic points. An isometric C*-algebra homomorphism of the quotient C*-algebra \({{\mathfrak {B}}^\pi={\mathfrak {B}}/{\mathcal {K}}}\), where \({{\mathcal {K}}}\) is the ideal of compact operators, into an n × n matrix algebra associated to a C*-algebra \({{\mathfrak {B}}_0}\) of singular integral operators with shifts having only fixed points is established making use of a spectral measure. Based on this generalization of the Litvinchuk–Gohberg–Krupnik reduction scheme, a symbol calculus for the C*-algebra \({{\mathfrak {B}}}\) as well as a Fredholm criterion for the operators in \({{\mathfrak {B}}}\) are obtained.

On the spectrum of the algebra of singular integral operators with discontinuities in symbols in momenta and coordinates

We study the C*-algebra generated in $ {L^2}\left( \mathbb{R} \right) $ by operators of multiplication by functions with finitely many discontinuities of the first kind and by convolution operators with the Fourier transforms of such functions. The algebra is represented as the restricted direct sum . We express the spectrum of the restricted direct sum in terms of the spectra of its summands. This result is used to express the spectrum of the algebra in terms of the spectra of and . We describe all equivalence classes of irreducible representations of the algebra , the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra by the ideal of compact operators and by the ideal com generated by the commutators of elements of the algebra . Bibliography: 14 titles.

C*-Algebras of Singular Integral Operators with Shifts Having the Same Nonempty Set of Fixed Points

The C*-subalgebra \({\mathfrak{B}}\) of \({\mathcal{B}}(L^2({\mathbb{T}}))\) generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms \(g : {\mathbb{T}} \rightarrow {\mathbb{T}}\) with any nonempty set of common fixed points is studied. A symbol calculus for the C*-algebra \({\mathfrak{B}}\) and a Fredholm criterion for its elements are obtained. For the C*-algebra \(\mathcal{A}\) composed by all functional operators in \({\mathfrak{B}}\), an invertibility criterion for its elements is also established. Both the C*-algebras \({\mathfrak{B}}\) and \({\mathcal{A}}\) are investigated by using a generalization of the local-trajectory method for C*-algebras associated with C*-dynamical systems which is based on the notion of spectral measure.

Inversion in an algebra of singular integral operators of generalized homogeneity

A result of M. Christ on inversion of translation-invariant homogeneous singular integral operators on a Lie group having kernels with Sobolev-type smoothness is shown to hold without the homogeneity assumption. Instead we impose uniform size and smoothness conditions on the dilates of the kernel over a fixed annulus. These are analogous to the Hormander condition for multipliers. Using natural embeddings between Sobolev and Lipschitz spaces we show that this generalization gives an inversion result for singular integral operators satisfying the standard Calderon-Zygmund Lipschitz estimates. Applications to the algebra of operators bounded on a Banach space of molecules are also given.

Semi-Fredholm singular integral operators with piecewise continuous coefficients on weighted variable Lebesgue spaces are Fredholm

Suppose $\Gamma$ is a Carleson Jordan curve with logarithmic whirl points, $\varrho$ is a Khvedelidze weight, $p:\Gamma\to(1,\infty)$ is a continuous function satisfying $|p(\tau)-p(t)|\le -\mathrm{const}/\log|\tau-t|$ for $|\tau-t|\le 1/2$, and $L^{p(\cdot)}(\Gamma,\varrho)$ is a weighted generalized Lebesgue space with variable exponent. We prove that all semi-Fredholm operators in the algebra of singular integral operators with $N\times N$ matrix piecewise continuous coefficients are Fredholm on $L_N^{p(\cdot)}(\Gamma,\varrho)$.

Spectral measures in [formula omitted]-algebras of singular integral operators with shifts

The C ∗ -subalgebra B of B ( L 2 ( T ) ) generated by all multiplication operators by slowly oscillating and piecewise continuous functions, by the Cauchy singular integral operator and by the range of a unitary representation of an amenable group of diffeomorphisms which acts topologically freely on T is studied. A symbol calculus for the C ∗ -algebra B and a Fredholm criterion for the operators B ∈ B are established by using a generalization of the local-trajectory method for studying C ∗ -algebras associated with C ∗ -dynamical systems. This method is related to the Allan–Douglas local principle and its generalization is based on the notion of spectral measure.

On algebras of singular integral operators with shift in Hölder weighted spaces

We consider algebras of singular integral operators with shift and piecewise Holder coefficients in a Holder weighted space on a Lyapunov contour. For this algebra, we construct the similarity isomorphism to the algebra of singular integral operators with piecewise Holder coefficients in a Holder space with “canonical” weight on the circle. We construct the symbol calculus, formulate necessary and sufficient conditions for the Fredholm property, and give the formula for the index of Fredholm operators.

Fredholmness of singular integral operators with discrete subexponentialgroups of shifts on Lebesgue spaces

AbstractThe paper is devoted to an application of a general local method of studying the Fredholmness of nonlocal bounded linear operators to Banach algebras of singular integral operators with piecewise continuous coefficients and discrete subexponential groups of piecewise smooth shifts acting topologically freely on composed contours. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Singular Integral Operators with Fixed Singularities on Weighted Lebesgue Spaces

The paper is devoted to study of singular integral operators with fixed singularities at endpoints of contours on weighted Lebesgue spaces with general Muckenhoupt weights. Compactness of certain integral operators with fixed singularities is established. The membership of singular integral operators with fixed singularities to Banach algebras of singular integral operators on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights is proved on the basis of Balakrishnan’s formula from the theory of strongly continuous semi-groups of closed linear operators. Symbol calculus for such operators, Fredholm criteria and index formulas are obtained.

The [formula omitted]-algebra of singular integral operators with semi-almost periodic coefficients

We establish a Fredholm criterion for the operators belonging to the C ∗ -algebra generated by singular integral operators with semi-almost periodic matrix coefficients. The result is applied to Toeplitz-like operators that are perturbed by integral operators with fixed singularities at infinity, in which case it leads to an effectively verifiable Fredholm criterion together with an index formula. Our approach is based on an isomorphism theorem for C ∗ -algebras associated with C ∗ -dynamical systems and the notion of canonical generalized AP factorization for almost periodic matrix functions.

C * -algebras of singular integral operators in domains with oscillating conical singularities

For a domain \(\) with singular points on the boundary, we consider a C*-algebra \(\) of operators acting in the weighted space \(\) . It is generated by the operators of multiplication by continuous functions on \(\) and the operators \(\) where σ is a homogeneous function. We show that the techniques of limit operators apply to define a symbol algebra for \(\) . When combined with the local principle, this leads to describing the Fredholm operators in \(\) .

A shift-invariant algebra of singular integral operators with oscillating coefficients

LetB be the Banach algebra of all bounded linear operators on the weighted Lebesgue spaceL p (T, ω) with an arbitrary Muckenhoupt weight ω on the unit circleT, and $$\mathfrak{A}$$ the Banach subalgebra ofB generated by the operators of multiplication by piecewise continuous coefficients and the operatorse h,λS T e −1 I (h∈R, λ∈T) whereS T is the Cauchy singular integral operator ande h,λ(t)=exp(h(t+λ)/(t−λ)),t∈T. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra $$\mathfrak{A}$$ and its matrix analogue $$\mathfrak{A}_{NxN} $$ . These shift-invariant algebras arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms ofT onto itself with second Taylor derivatives.

Algebra of singular integral operators with a Carleman backward slowly oscillating shift

In this paper we study the Banach algebra of singular integral operators with piecewise continuous coefficients and a Carleman orientation-reversing slowly oscillating shift on the Lebesgue space with a power weight on the unit circle. The slow oscillation of the shift derivative, in contrast to the classic assumption on its piecewise continuity, leads to the appearance of massive local spectra for the considered operators. Applying localization techniques and the theory of Mellin pseudodifferential and associated limit operators, we construct a symbol calculus for the above-mentioned operator algebra and find a Fredholm criterion and an index formula for the operators in this algebra in terms of their symbols.

Local Spectra and Index of Singular Integral Operators with Piecewise Continuous Coefficients on Composed Curves

AbstractWe establish a symbol calculus for deciding whether singular integral operators with piecewise continuous coefficients are Fredholm on the Lebesgue space Lp(Γ W) where 1 &lt; p &lt; ∞, Γ is a composed Carleson curve, and W is a Muckenhoupt weight in the class Ap(Γ). We also provide index formulas for the operators in the closed algebra of singular integral operators with piecewise continuous matrix‐valued coefficients. Our main theorem is based upon three pillars: on the identification of the local spectrum of the Cauchy singular integral operator at the endpoints of simple Carleson arcs, on an appropriate “N projections theorem”, and on results of geometric function theory pertaining to the problem of extending Carleson curves and Muckenhoupt weights.

$C^\ast$-Algebras of singular integral operators and Toeplitz operators associated with measure-preserving transformations

We show in the setting of measure-theoretical dynamical systems that certain singular integral operators and Toeplitz operators with the same coefficients but represented on different $L^2$-spaces are $C^\ast $-algebraically equivalent.