Fredholm Symbol Research Articles

C*-ALGEBRAS OF POLY-BERGMAN TYPE OPERATORS WITH PIECEWISE SLOWLY OSCILLATING COEFFICIENTS

Given \(n,m\in \mathbb {N}\) and a simply connected uniform domain \(U\subset \mathbb {C}\) with a sufficiently smooth boundary \(\partial U\), we study the \(C^{\ast}\)-algebra $$\begin{aligned} {\mathfrak B}_{U,n,m}({\mathfrak L}):=\mathrm{alg}\{aI,B_{U,1},\ldots ,B_{U,n},\widetilde{B}_{U,1},\ldots ,\widetilde{B}_{U,m}: a\in {\mathfrak X}({\mathfrak L})\}, \end{aligned}$$generated by the operators of multiplication by functions in \({\mathfrak X}({\mathfrak L})\), by the poly-Bergman projections \(B_{U,1},\ldots ,B_{U,n}\) and by the anti-poly-Bergman projections \(\widetilde{B}_{U,1},\ldots ,\widetilde{B}_{U,m}\) acting on the Lebesgue space \(L^2(U)\). The \(C^{\ast}\)-algebra \({\mathfrak X}({\mathfrak L})\) is generated by the set \(SO_{\partial} (U)\) of all bounded continuous functions on U that slowly oscillate at points of \(\partial U\) and by the set \(PC({\mathfrak L})\) of all piecewise continuous functions on the closure \(\overline{U}\) of U with discontinuities on a finite union \({\mathfrak L}\) of piecewise Dini-smooth curves that have one-sided tangents at every point \(z\in {\mathfrak L}\), possess a finite set \(Y={\mathfrak L}\cap \partial U\), do not form cusps, and are not tangent to \(\partial U\) at the points \(z\in Y\). Making use of the Allan-Douglas local principle, the limit operators techniques, quasicontinuous maps, and properties of \(SO_{\partial} (U)\) functions, a Fredholm symbol calculus for the \(C^{\ast}\)-algebra \({\mathfrak B}_{U,n,m}({\mathfrak L})\) is constructed and a Fredholm criterion for its operators is obtained.

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.

On Mellin pseudodifferential operators with quasicontinuous symbols

Let be the C∗‐algebra of quasicontinuous functions on and be the Cauchy singular integral operator, and let p ∈ (1, ∞). The paper is devoted to studying Mellin pseudodifferential operators with quasicontinuous ‐valued symbols on Lebesgue spaces , where is the Banach algebra of absolutely continuous functions of bounded total variation on the real line , and dμ(r) = dr/r. Applying obtained results on Mellin pseudodifferential operators with quasicontinuous ‐valued symbols, we study a Banach algebra of modified singular integral operators on the space , which is generated by the operators of the form where , , Rβ for is a singular integral operator with point singularities at 0 and ∞, Vα is the shift operator given by Vαf = f ∘ α, and α is a homeomorphism of [0, ∞] onto itself with . A Fredholm symbol calculus for the Banach algebra is constructed and a Fredholm criterion for the operators is established.

C*-algebras of Bergman Type Operators with Piecewise Slowly Oscillating Coefficients over Domains with Dini-Smooth Corners

Given a simply connected domain $$U\subset \mathbb {C}$$ with a piecewise Dini-smooth boundary $$\partial U$$ which admits a finite set of Dini-smooth corners of openings lying in $$(0,2\pi ]$$ , we study the $$C^*$$ -algebra $$\mathfrak {B}_U=\{aI, B_U, \widetilde{B}_U:a\in \mathfrak {X}(\mathfrak {L})\}$$ generated by the operators of multiplication by functions in $$\mathfrak {X}(\mathfrak {L})$$ , and by the Bergman projection $$B_U$$ and anti-Bergman projection $$\widetilde{B}_U$$ acting on the Lebesgue space $$L^2(U)$$ . The $$C^*$$ -algebra $$\mathfrak {X}(\mathfrak {L})$$ is generated by all piecewise continuous functions on the closure $$\overline{U}$$ of U with discontinuities on a finite union $$\mathfrak {L}$$ of piecewise Dini-smooth curves that have one-sided tangents at every point, do not form cusps and are not tangent to $$\partial U$$ at the points of $$\mathfrak {L}\cap \partial U$$ , and by all bounded continuous functions on U that slowly oscillate at points of $$\partial U$$ . Making use of the Allan-Douglas local principle, the limit operators techniques and the Kehe Zhu results on the class $$Q=VMO_\partial (\mathbb {D}) \cap L^\infty (\mathbb {D})$$ , a Fredholm symbol calculus for the $$C^*$$ -algebra $$\mathfrak {B}_U$$ is constructed and a Fredholm criterion for the operators $$A\in \mathfrak {B}_U$$ is obtained.

C⁎-algebra of nonlocal convolution type operators

The C⁎-subalgebra B of all bounded linear operators on the space L2(R), which is generated by all multiplication operators by piecewise slowly oscillating functions, by all convolution operators with piecewise slowly oscillating symbols and by the range of a unitary representation of the group of all affine mappings on R, is studied. A faithful representation of the quotient C⁎-algebra Bπ=B/K in a Hilbert space, where K is the ideal of compact operators on the space L2(R), is constructed by applying an appropriate spectral measure decompositions, a local-trajectory method and the Fredholm symbol calculus for the C⁎-algebra of convolution type operators without shifts. This gives a Fredholm symbol calculus for the C⁎-algebra B and a Fredholm criterion for the operators B∈B.

$$C^*$$ C ∗ -algebras of Bergman Type Operators with Piecewise Continuous Coefficients Over Domains with Dini-Smooth Corners

For any simply connected domain \(U\subset \mathbb {C}\) with a piecewise Dini-smooth boundary \(\partial U\) which admits Dini-smooth corners of openings lying in \((0,2\pi ]\), the \(C^*\)-algebra \({\mathfrak B}_U=\mathrm{alg}\{aI,B_U,\widetilde{B}_U:a\in PC({\mathfrak L})\}\) generated by the operators of multiplication by piecewise continuous functions with discontinuities on a finite union \({\mathfrak L}\subset U\) of piecewise Dini-smooth curves that have one-sided tangents at every point, do not form cusps and are not tangent to \(\partial U\) at the points of \({\mathfrak L}\cap \partial U\), and by the Bergman projection \(B_U\) and anti-Bergman projection \(\widetilde{B}_U\) acting on the Lebesgue space \(L^2(U)\) is studied. A Fredholm symbol calculus for the \(C^*\)-algebra \({\mathfrak B}_U\) is constructed and a Fredholm criterion for the operators \(A\in {\mathfrak B}_U\) in terms of their symbols is established. Results essentially depend on openings of corners and angles between curves of discontinuity of coefficients at the points \(z\in \partial U\cap {\mathfrak L}\).

Algebras of Convolution-Type Operators with Piecewise Slowly Oscillating Data on Weighted Lebesgue Spaces

Let \({\mathcal B}_{p,w}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \(L^p(\mathbb {R},w)\), where \(p\in (1,\infty )\) and w is a Muckenhoupt weight. We study the Banach subalgebra \(\mathfrak {A}_{p,w}\) of \({\mathcal B}_{p,w}\) generated by all multiplication operators aI (\(a\in \mathrm{PSO}^\diamond \)) and all convolution operators \(W^0(b)\) (\(b\in \mathrm{PSO}_{p,w}^\diamond \)), where \(\mathrm{PSO}^\diamond \subset L^\infty (\mathbb {R})\) and \(\mathrm{PSO}_{p,w}^\diamond \subset M_{p,w}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \(\mathbb {R}\cup \{\infty \}\), and \(M_{p,w}\) is the Banach algebra of Fourier multipliers on \(L^p(\mathbb {R},w)\). For any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra \({\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}\) generated by the operators \(aW^0(b)\) with slowly oscillating data \(a\in \mathrm{SO}^\diamond \) and \(b\in \mathrm{SO}^\diamond _{p,w}\). Then, under some condition on the weight w, we complete constructing a Fredholm symbol calculus for the Banach algebra \(\mathfrak {A}_{p,w}\) in comparison with Karlovich and Loreto Hernandez (Integr. Equations Oper. Theory 74:377–415, 2012) and Karlovich and Loreto Hernandez (Integr. Equations Oper. Theory 75:49–86, 2013) and establish a Fredholm criterion for the operators \(A\in \mathfrak {A}_{p,w}\) in terms of their symbols. A new approach to determine local spectra is found.

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.

TOEPLITZ OPERATORS AND THE ESSENTIAL BOUNDARY ON POLYANALYTIC FUNCTIONS

Let j be a nonzero integer and let U be a bounded domain. We construct a Fredholm symbol calculus for the C*-algebra generated by the poly-Bergman projection and the operators of multiplication by continuous functions. We define the j-removal boundary in Hilbert spaces of polyanalytic functions and prove that the quotient poly-Toeplitz C*-algebra generated by cosets of poly-Toeplitz operators with continuous symbols is *-isomorphic to the C*-algebra of continuous functions over the j-essential boundary. Unlike the Bergman case, we also show that if j ≠ ±1 then the j-essential boundary coincides with the set of non-isolated points on the boundary.

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

Let \({\mathcal{B}_{p,w}}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \({L^p(\mathbb{R},w)}\) , where \({p\in(1,\infty)}\) and w is a Muckenhoupt weight. We study the Banach subalgebra \({\mathfrak{U}_{p,w}}\) of \({\mathcal{B}_{p,w}}\) generated by all multiplication operators aI (\({a\in PSO^\diamond}\)) and all convolution operators W0(b) (\({b\in PSO_{p,w}^\diamond}\)), where \({PSO^\diamond\subset L^\infty(\mathbb{R})}\) and \({PSO_{p,w}^\diamond\subset M_{p,w}}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \({\mathbb{R}\cup\{\infty\}}\) , and Mp,w is the Banach algebra of Fourier multipliers on \({L^p(\mathbb{R},w)}\) . Under some conditions on the Muckenhoupt weight w, using results of the local study of \({\mathfrak{U}_{p,w}}\) obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra \({\mathfrak{U}_{p,w}}\) and establish a Fredholm criterion for the operators \({A\in\mathfrak{U}_{p,w}}\) in terms of their Fredholm symbols. In four partial cases we obtain for \({\mathfrak{U}_{p,w}}\) more effective results.

Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. I: Local and Structural Study

Let \({\mathcal{B}_{p,w}}\) be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space \({L^{p}(\mathbb{R}, w)}\), where \({p \in (1, \infty)}\) and w is a Muckenhoupt weight. We study the Banach subalgebra \({\mathfrak{A}_{p,w}}\) of \({\mathcal{B}_{p,w}}\) generated by all multiplication operators aI (\({a \in PSO^{\diamond}}\)) and all convolution operators W0(b) (\({b \in PSO_{p,w}^{\diamond}}\)), where \({PSO^{\diamond} \subset L^{\infty}(\mathbb{R})}\) and \({PSO_{p,w}^{\diamond} \subset M_{p,w}}\) are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of \({\mathbb{R} \cup \{\infty\}}\), and Mp,w is the Banach algebra of Fourier multipliers on \({L^{p}(\mathbb{R}, w)}\). Under some conditions on the Muckenhoupt weight w, we construct a Fredholm symbol calculus for the Banach algebra \({\mathfrak{A}_{p,w}}\) and establish a Fredholm criterion for the operators \({A \in \mathfrak{A}_{p,w}}\) in terms of their Fredholm symbols. To study the Banach algebra \({\mathfrak{A}_{p,w}}\) we apply the theory of Mellin pseudodifferential operators, the Allan–Douglas local principle, the two idempotents theorem and the method of limit operators. The paper is divided in two parts. The first part deals with the local study of \({\mathfrak{A}_{p,w}}\) and necessary tools for studying local algebras.

On the algebras generated by Toeplitz operators with piecewise continuous symbols

We explain an apparent disagreement between the fact that the Fredholm symbol algebras of two different C∗-algebras generated by Toeplitz operators with piecewise continuous symbols, acting on the Hardy space and the Bergman space, have the same Fredholm symbol algebras and the same symbol homomorphism on generating operator, and the fact that the initial generators of these algebras are not unitary equivalent modulo compact operators.

On the algebras generated by Toeplitz operators with piecewise continuous symbols

We explain an apparent disagreement between the fact that the Fredholm symbol algebras of two different C∗-algebras generated by Toeplitz operators with piecewise continuous symbols, acting on the Hardy space and the Bergman space, have the same Fredholm symbol algebras and the same symbol homomorphism on generating operator, and the fact that the initial generators of these algebras are not unitary equivalent modulo compact operators.

C*-algebras of Bergman-type operators with piecewise continuous coefficients and shifts

Let U be a bounded simply connected domain in ℂ with sufficiently smooth boundary Γ. Let G be a commutative group of conformal mappings of onto itself which is similar to the group of elliptic, hyperbolic or parabolic mappings of the closed unit disc onto itself, and let be a G-invariant set of simple Lyapunov curves in such that for every an at most finite number of curves in are intersecting at z and at every z ∈ Γ these curves form with Γ pairwise distinct angles lying in (0, π). Let be the C*-algebra generated by n poly-Bergman projections, m anti-poly-Bergman projections and by all multiplication operators aI acting on the space L 2(U), where a are piecewise continuous functions on with possible discontinuities on subsets of that are continuous at common fixed points of g ∈ G. For mentioned groups G, applying a local-trajectory method and a Fredholm symbol calculus for the C*-algebra , we establish Fredholm criteria for the operators B in the C*-algebras generated by all operators and all weighted shift operators W g (g ∈ G), where W g f = g′(f ○ g) for f ∈ L 2(U).

Affine coherent states and Toeplitz operators

We study a parameterized family of Toeplitz operators in the context of affine coherent states based on the Calderón reproducing formula (= resolution of unity on ) and the specific admissible wavelets (= affine coherent states in ) related to Laguerre functions. Symbols of such Calderón–Toeplitz operators as individual coordinates of the affine group (= upper half-plane with the hyperbolic geometry) are considered. In this case, a certain class of pseudo-differential operators, their properties and their operator algebras are investigated. As a result of this study, the Fredholm symbol algebras of the Calderón–Toeplitz operator algebras for these particular cases of symbols are described.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S * related to the Carleson–Hunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in \({L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}\) and \({\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}\) on the weighted Lebesgue spaces \({L^p(\mathbb{R},w)}\) , with 1 < p < ∞ and \({w\in A_p(\mathbb{R})}\) . The Banach algebras \({L^\infty(\mathbb{R}, V(\mathbb{R}))}\) and \({PC(\overline{\mathbb{R}}, V(\mathbb{R}))}\) consist, respectively, of all bounded measurable or piecewise continuous \({V(\mathbb{R})}\) -valued functions on \({\mathbb{R}}\) where \({V(\mathbb{R})}\) is the Banach algebra of all functions on \({\mathbb{R}}\) of bounded total variation, and the Banach algebra \({\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}\) consists of all Lipschitz \({V_d(\mathbb{R})}\) -valued functions of exponent \({\gamma \in (0,1]}\) on \({\mathbb{R}}\) where \({V_d(\mathbb{R})}\) is the Banach algebra of all functions on \({\mathbb{R}}\) of bounded variation on dyadic shells. Finally, for the Banach algebra \({\mathfrak{A}_{p,w}}\) generated by all pseudodifferential operators a(x, D) with symbols \({a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}\) on the space \({L^p(\mathbb{R}, w)}\) , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators \({A \in \mathfrak{A}_{p,w}}\) .

AN ALGEBRA OF PSEUDODIFFERENTIAL OPERATORS WITH SLOWLY OSCILLATING SYMBOLS

Let $V(\mathbb{R})$ denote the Banach algebra of absolutely continuous functions of bounded total variation on $\mathbb{R}$, and let $\mathcal{B}_p$ be the Banach algebra of bounded linear operators acting on the Lebesgue space $L^p(\mathbb{R})$ for $1 &lt; p &lt; \infty$. We study the Banach algebra $\mathfrak{A}\subset\mathcal{B}_p$ generated by the pseudodifferential operators of zero order with slowly oscillating $V(\mathbb{R})$-valued symbols on $\mathbb{R}$. Boundedness and compactness conditions for pseudodifferential operators with symbols in $L^\infty (\mathbb{R}, V(\mathbb{R}))$ are obtained. A symbol calculus for the non-closed algebra of pseudodifferential operators with slowly oscillating $V(\mathbb{R})$-valued symbols is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. As a result, the quotient Banach algebra $\mathfrak{A}^\pi = {\mathfrak A} / \mathcal{K}$, where $\mathcal{K}$ is the ideal of compact operators in $\mathcal{B}_p$, is commutative and involutive. An isomorphism between the quotient Banach algebra $\mathfrak{A}^\pi$ of pseudodifferential operators and the Banach algebra $\widehat{\mathfrak{A}}$ of their Fredholm symbols is established. A Fredholm criterion and an index formula for the pseudodifferential operators $A \in \mathfrak{A}$ are obtained in terms of their Fredholm symbols.

On algebras of two dimensional singular integral operators with homogeneous discontinuities in symbols

We describe the Fredholm symbol algebra for theC*-algebra generated by two dimensional singular integral operators, acting onL2(ℝ2), 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.

Properties of generalized Toeplitz operators

An operatorX:\(X:\mathcal{H}_1 \to \mathcal{H}_2 \) is said to be a generalized Toeplitz operator with respect to given contractionsT1 andT2 ifX=T2XT1*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foias, and Ptak and Vrbova, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.

Pseudodifferential operators with heavy spectrum

We establish Fredholm criteria and index formulas for one-dimensional zero-order pseudodifferential operators with piecewise continuous generating functions onLp spaces with Muckenhoupt weights. The Fredholm symbol of such operators is shown to be a matrix function defined on a set which, roughly speaking, is a cylinder with a certain collection of horn shaped handles. The presence of these horns implies that, unlike the case ofLp spaces without weight or with so-called power weights, the spectrum may contain heavy parts, i. e. the set of the interior points of the spectrum need not be empty. Our proof makes essential use of recent results by Finck, Roch, Silbermann, Gohberg, and Krupnik on the inverse closedness of certain Banach algebras.