C * -algebras of singular integral operators in domains with oscillating conical singularities (original) (raw)
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Integral Equations and Operator Theory, 2001
We describe the Fredholm symbol algebra for the C*-al~ebra generated by two dimensional singular integral operators, acting on L2(]R ), and whose symbols admit homogeneous discontinuities. Locally these discontinuities are modeled by homogeneous functions having slowly oscillating (and, in particular, piecewise continuous) discontinuities on a system of rays outgoing from the origin. These results extend the well-known Plamenevsky results for the two dimensional case. We present here an alternative and much clearer approach to the problem.
Journal of Mathematical Sciences, 2011
We study the C * -algebra B generated in L 2 (R) 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 B is represented as the restricted direct sum A 1 ⊕ C A 2 . 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 B in terms of the spectra of A 1 and A 2 . We describe all equivalence classes of irreducible representations of the algebra B, the topology on the spectrum of this algebra, and solving composition series. We discuss the abstract index group of the quotient algebra B by the ideal of compact operators and by the ideal com B generated by the commutators of elements of the algebra B. Bibliography: 14 titles.
An algebra of integral operators with fixed singularities in kernels
Integral Equations and Operator Theory, 1999
We continue the study of algebras generated by the Cauchy singular integral operator and integral operators with fixed singularities on the unit interval, started in R.Duduehava, E.Shargorodsky, 1990. Such algebras emerge when one considers singular integral operators with complex conjugation on curves with cusps. As one of possible applications of the obtained results we find an explicit formula for the local norms of the Cauchy singular integral operator on the Lebesgue space L~(F, p) , where Y is a curve with cusps of arbitrary order and p is a power weight. For curves with angles and cusps of order 1 the formula was already known (see R.
Singular Integral Operators with Fixed Singularities on Weighted Lebesgue Spaces
Integral Equations and Operator Theory, 2004
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.
A -algebra of singular integral operators with shifts admitting distinct fixed points
Journal of Mathematical Analysis and Applications, 2014
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 U g 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.
C*-Algebras of Poly-Bergman Type Operators with Piecewise Slowly Oscillating Coefficients
Journal of Mathematical Sciences
Given n, m ∈ ℕ and a simply connected uniform domain U ⊂ ℂ with a sufficiently smooth boundary U , we study the C *-algebra generated by the operators of multiplication by functions in () , by the poly-Bergman projections B U,1 , … , B U,n and by the anti-poly-Bergman projections B U,1 , … ,B U,m acting on the Lebesgue space L 2 (U). The C *-algebra () is generated by the set SO (U) of all bounded continuous functions on U that slowly oscillate at points of U and by the set PC() of all piecewise continuous functions on the closure U of U with discontinuities on a finite union of piecewise Dini-smooth curves that have one-sided tangents at every point z ∈ , possess a finite set Y = ∩ U , do not form cusps, and are not tangent to U at the points z ∈ Y. Making use of the Allan-Douglas local principle, the limit operators techniques, quasicontinuous maps, and properties of SO (U) functions, a Fredholm symbol calculus for the C *-algebra U,n,m () is constructed and a Fredholm criterion for its operators is obtained.
C*-Algebras of Singular Integral Operators with Shifts Having the Same Nonempty Set of Fixed Points
Complex Analysis and Operator Theory, 2008
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 g : T → T with any nonempty set of common fixed points is studied. A symbol calculus for the C * -algebra B and a Fredholm criterion for its elements are obtained. For the C * -algebra A composed by all functional operators in B, an invertibility criterion for its elements is also established. Both the C * -algebras B and 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.
Spectral measures in -algebras of singular integral operators with shifts
Journal of Functional Analysis, 2007
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.