Piecewise Continuous Symbols Research Articles

Asymptotics of block Toeplitz determinants with piecewise continuous symbols

AbstractWe determine the asymptotics of the block Toeplitz determinants as for matrix‐valued piecewise continuous functions with a finitely many jumps under mild additional conditions. In particular, we prove that where , , and are constants that depend on the matrix symbol and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix‐valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.

Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators $T$ with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator $T$, that is, by the behavior of $\exp(-iTt) f$ for $t\to\pm\infty$. It turns out that the spectrum is naturally partitioned into three disjoint subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation properties are modeled by the continuous part of the symbol, whereas on the thin spectrum, the model operator is determined by the jumps of the symbol. On the mixed spectrum, these two types of the asymptotic evolution of $\exp(-iTt) f$ coexist. This classification is justified in the framework of scattering theory. We prove the existence of wave operators that relate the model operators with the Toeplitz operator $T$. The ranges of these wave operators are pairwise orthogonal, and their orthogonal sum exhausts the whole space, i.e., the set of these wave operators is asymptotically complete.

Kernels of Unbounded Toeplitz Operators and Factorization of Symbols

We consider kernels of unbounded Toeplitz operators in H^p({mathbb {C}}^{+}) in terms of a factorization of their symbols. We study the existence of a minimal Toeplitz kernel containing a given function in H^p({mathbb {C}}^{+}), we describe the kernels of Toeplitz operators whose symbol possesses a certain factorization involving two different Hardy spaces and we establish relations between the kernels of two operators whose symbols differ by a factor which corresponds, in the unit circle, to a non-integer power of z. We apply the results to describe the kernels of Toeplitz operators with non-vanishing piecewise continuous symbols.

On spectral analysis of self-adjoint Toeplitz operators

The paper pursues three objectives. First, we provide an expanded version of the spectral analysis of self-adjoint Toeplitz operators, initially built by M. Rosenblum in the 1960's. We offer some improvements to Rosenblum's approach: for instance, our proof of the absolute continuity, relying on a weak version of the limiting absorption principle, is more direct. Secondly, we study in detail Toeplitz operators with finite spectral multiplicity. In particular, we introduce generalized eigenfunctions and investigate their properties. Thirdly, we develop a more detailed spectral analysis for piecewise continuous symbols. This is necessary for construction of scattering theory for Toeplitz operators with such symbols.

The Spectral Density of Hankel Operators with Piecewise Continuous Symbols

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the Ntimes N truncated Hilbert matrix for large values of N. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g. square or triangular truncation).

On the Fredholm Property of a Class of Convolution-Type Operators

The notions of the L-convolution operator and the ℒ-Wiener–Hopf operator are introduced by replacing the Fourier transform in the definition of the convolution operator by a spectral transformation of the self-adjoint Sturm–Liouville operator on the axis ℒ. In the case of the zero potential, the introduced operators coincide with the convolution operator and theWiener–Hopf integral operator, respectively. A connection between the ℒ-Wiener–Hopf operator and singular integral operators is revealed. In the case of a piecewise continuous symbol, a criterion for the Fredholm property and a formula for the index of the ℒ-Wiener–Hopf operator in terms of the symbol and the elements of the scattering matrix of the operator ℒ are obtained.

On a Class of L-Wiener-Hopf Operators

By replacement in the definition of the convolution operator of Fourier transform by a spectral transform of a selfadjoint Sturm-Liouville operator on the axis L, the concepts of Lconvolution and L-Wiener-Hopf operators are introduced. The case of the reflectorless potentials with a single eigenvalue is considered. A relationship between the Wiener-Hopf and L-Wiener- Hopf operators is established. In the case of piecewise continuous symbol the Fredholm property and invertibility of the L-Wiener-Hopf operator are investigated.

Spectral and scattering theory of self-adjoint Hankel operators with piecewise continuous symbols

We develop the spectral and scattering theory for self-adjoint Hankel operators $H$ with piecewise continuous symbols. In this case every jump of the symbol gives rise to a band of the absolutely continuous spectrum of $H$. We construct wave operators relating simple model (that is, explicitly diagonalizable) Hankel operators for each jump and the given Hankel operator $H$. We show that the set of all these wave operators is asymptotically complete. This determines the absolutely continuous part of $H$. We also prove that the singular continuous spectrum of $H$ is empty and that its eigenvalues may accumulate only to thresholds in the absolutely continuous spectrum. All these results are reformulated in terms of Hankel operators realized as matrix or integral operators.

Finite section approximations in an algebra of convolution, multiplication and flip operators on [formula omitted

This paper is concerned with the approximation of solutions of operator equations using the finite sections method with the operators belonging to the closed subalgebra of B(Lp(R)), 1<p<∞, generated by operators of multiplication by piecewise continuous functions in Ṙ, operators of convolution by piecewise continuous Fourier multipliers and the flip operator. This algebra includes Wiener–Hopf and Hankel operators with piecewise continuous symbols. To prove the result, we use algebraic techniques and introduce a larger algebra of sequences, which contains the special sequences we are interested and the usual operator algebra generated by the operators of multiplication, convolution and flip. There is a direct relationship between the applicability of the finite section method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship and using local principles, we construct locally equivalent representations that allow to derive invertibility criteria.

Spectral theory of piecewise continuous functions of self-adjoint operators

Let $H_0$, $H$ be a pair of self-adjoint operators for which the standard assumptions of the smooth version of scattering theory hold true. We give an explicit description of the absolutely continuous spectrum of the operator $\mathcal{D}_\theta=\theta(H)-\theta(H_0)$ for piecewise continuous functions $\theta$. This description involves the scattering matrix for the pair $H_0$, $H$, evaluated at the discontinuities of $\theta$. We also prove that the singular continuous spectrum of $\mathcal{D}_\theta$ is empty and that the eigenvalues of this operator have finite multiplicities and may accumulate only to the "thresholds" of the absolutely continuous spectrum of $\mathcal{D}_\theta$. Our approach relies on the construction of "model" operators for each jump of the function $\theta$. These model operators are defined as certain symmetrised Hankel operators which admit explicit spectral analysis. We develop the multichannel scattering theory for the set of model operators and the operator $\theta(H)-\theta(H_0)$. As a by-product of our approach, we also construct the scattering theory for general symmetrised Hankel operators with piecewise continuous symbols.

On C*-Algebras of Toeplitz Operators on the Harmonic Bergman Space

Commutative algebras of Toeplitz operators acting on the Bergman space on the unit disk have been completely classified in terms of geometric properties of the symbol class. The question when two Toeplitz operators acting on the harmonic Bergman space commute is still open. In some papers, conditions on the symbols have been given in order to have commutativity of two Toeplitz operators. In this paper, we describe three different algebras of Toeplitz operators acting on the harmonic Bergman space: The C*-algebra generated by Toeplitz operators with radial symbols, in the elliptic case; the C*-algebra generated by Toeplitz operators with piecewise continuous symbols, in the parabolic and hyperbolic cases. We prove that the Calkin algebra of the first two algebras are commutative, like in the case of the Bergman space, while the last one is not.

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.

On the Structure of the C *-Algebra Generated by Toeplitz Operators with Piece-wise Continuous Symbols

We study the C ∗ -algebra generated by Toeplitz operators with piece-wise continuous symbols acting on the Bergman space A 2 (D )o n the unit disk D in C. We describe explicitly each operator from this algebra and characterize Toeplitz operators which belong to the algebra. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47C15.

An Approximation Theory for Operators Generated by Shifts

This paper develops a theory about the applicability of approximation methods to operators that can be represented as a function of a shift. An axiomatic approach is used, in which a small set of conditions that involve the operator and the method are proved to guarantee the applicability. All concrete methods we know for the underlying operators are subjected to this approach. We also study the behavior of the singular values and related questions. New results concerning Galerkin approximation methods for Mellin operators with piecewise continuous symbol are given to illustrate the application of the theory.

Automorphisms on the Toeplitz Algebra with Piecewise Continuous Symbol on the Unit Ball

In this paper, we obtain a Fredholm index formula for Toeplitz operators whose symbols are certain piecewise continuous function matrices on the unit ball. Moreover, using this formula, we discuss the automorphisms on the corresponding Toeplitz algebra

Approximation numbers for the finite sections of Toeplitz operators with piecewise continuous symbols

In this paper we discuss the asymptotic distribution of the approximation numbers of the finite sections for a Toeplitz operator T ( a ) ∈ L ( ℓ p ) , 1 < p < ∞ , where a is a piecewise continuous function on the unit circle. We prove that the behavior of the approximation numbers of the finite sections T n ( a ) = P n T ( a ) P n depends heavily on the Fredholm properties of the operators T ( a ) and T ( a ˜ ) ( a ˜ ( t ) = a ( 1 / t ) ) . In particular, if the operators T ( a ) and T ( a ˜ ) are Fredholm on ℓ p , then the approximation numbers of T n ( a ) have the so-called k-splitting property. But, in contrast with the case of continuous symbols, the splitting number k is in general larger than dim ker T ( a ) + dim ker T ( a ˜ ) .

On an Algebra of Toeplitz Operators With Piecewise Continuous Symbols

Let \(D = \{ z \in \;\mathbb{C}|\;|z| < 1\} \) and \(\mathcal{L}\) be a finite collection of smooth curves in D. Given k points \(z_1 , \ldots z_k \in \partial D\) consider the family \(M_r = M_r (z_1 , \ldots ,z_k ) \subset L_\infty (D)\) of all bounded and continuous functions on \(D\backslash \mathcal{L},\) with finite limits at \(z_1 , \ldots ,z_{k - 1} \) and radial limits at z k . We study the Toeplitz operator algebra \(\mathcal{T}(M_r )\) corresponding to M r and we prove that its Calkin algebra is isomorphic to the algebra of all continuous functions on some compact set. This fact implies that the commutator of two Toeplitz operators with this kind of symbols is compact. We also prove that the semi-commutator of such Toeplitz operators is not compact, in general.

Spline Approximation Methods with Uniform Meshes in Algebras of Multiplication and Convolution Operators

The purpose of this paper is to obtain necessary and sufficient conditions for maximum defect spline approximation methods with uniform meshes to be stable. The methods are applied to operators belonging to the closed subalgebra of ℒ (L2 (ℝ)) generated by operators of multiplication by piecewise continuous functions on ℝ and convolution operators also with piecewise continuousgenerating functions. To that purpose, a C*-algebra of sequences is introduced, which contains the special sequences of approximating operators we are interested in. There is a direct relationship between the applicability of the approximation method to a given operator and invertibility of the corresponding sequence in this C*-algebra. Exploring this relationship, applicability criteria are derived by the use of C*-algebra and Banach algebra techniques (essentialization, localization andidentification of the local algebras by means of construction of locally equivalent representations). Finally, examples are presented, including explicit conditions for the applicability of spline Galerkin methods to Wiener-Hopf operators with piecewise continuous symbols.

Automorphisms of the Toeplitz algebra with piecewise continuous symbol

The automorphism group of the Toeplitz algebra generated by the Toeplitz operators, whose symbols are continuous functions on the circle beside finitely fixedpoints, is characterized.