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 ℒ︁ (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.     

A fredholm study for convolution operators with piecewise continuous symbols on a union of a finite and a semi-infinite intervel 1

We study convolution type operators with kernels that have Fourier transforms in the class of piecewise continuous matrix functions. These convolution operators are assumed to act between Sobolev spaces defined on a union of a finite and a semi-infinite intervel. The main result is a criterion for the Fredholm property of these operators. An application to a problem related to diffraction theory is illustrated.     

Completely bounded disjointness preserving operators between Fourier algebras

In this paper, we characterize surjective completely bounded disjointness preserving linear operators on Fourier algebras of locally compact amenable groups. We show that such operators are given by weighted homomorphisms induced by piecewise affine proper maps.     

Algebras of Singular Integral Operators with Piecewise Continuous Coefficients on Reflexive Orlicz Spaces

We consider singular integral operators with piecewise continuous coefficients on reflexive Orlicz spaces L_m(σ) which are generalizations of the Lebesgue spaces L_P(σ), 1 < p < ∞. We suppose that σ belongs to a large class of Carleson curves, including curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. For the singular integral operator associated with the Riemann boundary value problem with a piecewise continuous coefficient G, we establish a Fredholm criterion and an index formula in terms of the essential range of G complemented by spiralic horns depending on the Boyd indices of L_M(σ) and contour properties. Our main result is a symbol calculus for the closed algebra of singular integral operators with piecewise continuous matrix - valued coefficients on L_Mⁿ(σ).     

有界多连通域的具分段连续符号的Toeplitz算子

探讨多连通域的Bergman空间上的具有分段连续符号的Toeplitz算子,刻画了它们的本质谱和Fredholm指标.     

Transport equations and perturbations of boundary conditions

We provide a new perturbation theorem for substochastic semigroups on abstract AL spaces extending Kato's perturbation theorem to nondensely defined operators. We show how it can be applied to piecewise deterministic Markov processes and transport equations with abstract boundary conditions. We give particular examples to illustrate our results.     

Local C~k──boundary Regularity of Integral Solution Operators of the ──equations on Open Set with Piecewise C~1──boundary

ＬｏｃａｌＣ~ｋ──ｂｏｕｎｄａｒｙＲｅｇｕｌａｒｉｔｙｏｆＩｎｔｅｇｒａｌＳｏｌｕｔｉｏｎＯｐｅｒａｔｏｒｓｏｆｔｈｅ ──ｅｑｕａｔｉｏｎｓｏｎＯｐｅｎＳｅｔｗｉｔｈＰｉｅｃｅｗｉｓｅＣ~１──ｂｏｕｎｄａｒｙ￥ＭａＺｈｏｎｇｔａｉ（Ｄｅｐｔ．Ｏ?..     

Piecewise linear least squares approximations of Frobenius-Perron operators

We propose a piecewise linear numerical method based on least squares approximations for computing stationary density functions of Frobenius-Perron operators associated with piecewise C² and stretching mappings of the unit interval. We prove the weak convergence of the method for a class of Frobenius-Perron operators, and the numerical results show that it is also norm convergent and has a better convergence rate than the piecewise linear Markov approximation method.     

Unitary equivalence of one-parameter groups of Toeplitz and composition operators

The composition operators on H² whose symbols are hyperbolic automorphisms of the unit disk fixing ±1 comprise a one-parameter group and the analytic Toeplitz operators coming from covering maps of annuli centered at the origin whose radii are reciprocals also form a one-parameter group. Using the eigenvectors of the composition operators and of the adjoints of the Toeplitz operators, a direct unitary equivalence is found between the restrictions to zH² of the group of Toeplitz operators and the group of adjoints of these composition operators. On the other hand, it is shown that there is not a unitary equivalence of the groups of Toeplitz operators and the adjoints of the composition operators on the whole of H², but there is a similarity between them.     

Fredholm Operators and Riesz Theory for Polynomially Compact Operators

In this article, we give a characterization of a class of bounded Fredholm operators on a Banach space which is developed to present some general existence results of the operators equation of the second kind. The obtained results are used to describe the Riesz–Schauder theory of compact operators in the more general setting of polynomially compact operators.     

Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities

In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix characteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t ⁻¹, were found. Conditions for the invertibility of the matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Submitted: June 15, 2007. Revised: October 25, 2007. Accepted: November 5, 2007.     

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

We consider algebras of singular integral operators with shift and piecewise Hölder coefficients in a Hölder weighted space on a Lyapunov contour. For this algebra, we construct the similarity isomorphism to the algebra of singular integral operators with piecewise Hölder coefficients in a Hölder 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.Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 9, Suzdal Conference-3, 2003.     

Essentially subnormal operators

An operator is essentially subnormal if its image in the Calkin algebra is subnormal. We shall characterize the essentially subnormal operators as those operators with an essentially normal extension. In fact, it is shown that an essentially subnormal operator has an extension of the form ``normal plus compact'.

The essential normal spectrum is defined and is used to characterize the essential isometries. It is shown that every essentially subnormal operator may be decomposed as the direct sum of a subnormal operator and some irreducible essentially subnormal operators. An essential version of Putnam's Inequality is proven for these operators. Also, it is shown that essential normality is a similarity invariant within the class of essentially subnormal operators. The class of essentially hyponormal operators is also briefly discussed and several examples of essentially subnormal operators are given.

     

A Study of Probability Measures Through Commutators

The first-order moments and two families of commutators are proven to determine uniquely the moments of a probability measure on ℝ ^d . These families are the commutators between the annihilation and creation operators, and the commutators between the annihilation and preservation operators. An explicit method for recovering the moments from these commutators and first-order moments is presented.      

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

Toeplitz and composition operators on H2 (B n)

The composition operators with closed rangc on H²(B _n) are characterized, and the Fredholmness of products of Toeplitz and composition operators discussed. Moreover, using composition operators, the spectra of Toeplitz operators are studied. Project supported by the National Natural Sciencc Foundation of China and the Postdoctoral Science Foundation of China.     

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.Partially supported by CONaCYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.Partially supported by F. C. T. grant Praxis XXI/2/2.1/MAT/441/94, Portugal.     

Hardy空间上一类$m$-复对称Toeplitz算子

本文主要研究Toeplitz算子相对于一对置换的共轭算子是2-复对称的充要条件. 首先在经典的Hardy空间上介绍一类被称为一对置换的共轭算子, 其次完整地刻画了在这类共轭算子下Toeplitz算子是2-复对称的结构, 利用Toeplitz算子在Hardy空间的经典正规正交基下的矩阵表示来刻画2-复对称Toeplitz算子. 最后对于Toeplitz算子分别补充前提$f_n=-f_{-n}$和$f_n=f_{-n}$, 得到了更简化的结果. 在第二个前提下, 研究Toeplitz算子的3-复对称性, 得到$T_f$关于$C_{(i,j)}$是3-CSO的结果和是2-CSO相同.     

Symmetric anti-eigenvalue and symmetric anti-eigenvector

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.