Hankel-Toeplitz type operators onL a 1 (Ω)

This paper studies Hankel and Toeplitz operators on the Bergman spaceL _a ¹ () of bounded symmetric domains. These operators are defined in terms of a certain bounded projection onL ¹(,dV). The main results of the paper include several characterizations for the boundedness and (weak-star) compactness of these Hankel-Toeplitz type operators. When the symbol is conjugate holomorphic, our results here are similar to those obtained by Békollé, Berger, Coburn, and Zhu [2] in theL ²-Bergman space context.Research partially supported by the National Science Foundation     

On some properties of the positive m-Hessian operators in C 2(Ω)

We consider the p-Hessian operator T _p [u] := tr _p u _xx on the set ${\{u \in C^2(\bar{\Omega}), T_p[u] \geq \nu > 0, 1 \leq p \geq 0\}}$ and extend the common approach to the uniqueness and existence theorems. For instance, we prove the uniqueness of solutions to the Dirichlet problem for m-Hessian equations in ${C^2(\bar{\Omega})}$ for constant boundary data. We also give nonexistence theorems and, in addition, we establish the extremal properties of the functionals I _p [u; f] in ${C^2(\bar{\Omega})}$ and we derive some applications.     

The spectral bound of Schrödinger operators

Let V: R ^N [0, ] be a measurable function, and >0 be a parameter. We consider the behaviour of the spectral bound of the operator 1/2–V as a function of . In particular, we give a formula for the limiting value as , in terms of the integrals of V over subsets of R ^N on which the Laplacian with Dirichlet boundary conditions has prescribed values. We also consider the question whether this limiting value is attained for finite .     

Analytic semigroups generated in L 1(Ω) by second order elliptic operators via duality methods

Given an open domain (possibly unbounded) Ω?R ⁿ , we prove that uniformly elliptic second order differential operators, under nontangential boundary conditions, generate analytic semigroups in L ¹(Ω). We use a duality method, and, further, give estimates of first order derivatives for the resolvent and the semigroup, through properties of the generator in Sobolev spaces of negative order.     

Homomorphisms onC ∞ (E) andC ∞-bounding sets

We prove, for the class of real locally convex spacesE that are continuously and linearly injectable into somec ₀(), that every non-zero homomorphism on the algebraC (E) ofC -functions onE is given by a point evaluation at some point ofE. Furthermore, if every real-valuedC -function on the weak topology of a quasi-complete locally convex spaceE is bounded on a subsetA ofE, thenA is relatively weakly compact.     

On representations of Ψ* andC *-algebras ofb-pseudo-differential operators on manifolds with corners

The author computes the Jacobson topology and the corresponding Borel structure on the spectrum of . Here denotes the C^*-algebra generated by the algebra of b-pseudo-differential operators of orderO on a compact manifold with corners (in the sense of Melrose). The author also reduces the problem of classification of representations to the problem of classification of positive Borel measures on certain well-known Borel spaces. Bibliography:29 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 85–117.     

Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

Characterization of image spaces of Riemann-Liouville fractional integral operators on Sobolev spaces Wm,p (Ω)

Science China Mathematics - Fractional operators are widely used in mathematical models describing abnormal and nonlocal phenomena. Although there are extensive numerical methods for solving the...     

On the spectral properties of discrete Schrödinger operators

We use the method of the conjugate operator to prove a limiting absorption principle and the absence of the singular continuous spectrum for discrete Schrödinger operators. We also obtain local decay estimates. Our results apply to a large class of perturbating potentials V decaying arbitrarily slowly to zero at infinity.     

The extraordinary spectral properties of radially periodic Schrödinger operators

Since it became clear that the band structure of the spectrum of periodic Sturm-Liouville operatorst = - (d²/dr²) +q(r) does not survive a spherically symmetric extension to Schrödinger operatorsT =- Δ+ V with V(x) =q(¦x¦) for x ∈ ?^d,d ∈ ? 1, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum [μ₀, ∞[ ofT with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed Sturm-Liouville operatorst _c = t +(c/r ²). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues ofT more closely. An eigenvalue was discovered below the essential spectrum in the cased = 2, and it turned out that there are in fact infinitely many, accumulating at μ₀. Moreover, a method based on oscillation theory made it possible to count eigenvalues oft _c contributing to an interval of dense point spectrum ofT. We gained evidence that an asymptotic formula, valid forc → ∞, does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained.     

The order-preserving convergence for spectral approximation of self-adjoint completely continuous operators

This paper discusses the order-preserving convergence for spectral approximation of the self-adjoint completely continuous operator T.Under the condition that the approximate operator Th converges to T in norm,it is proven that the k-th eigenvalue of Th converges to the k-th eigenvalue of T.(We sorted the positive eigenvalues in decreasing order and negative eigenvalues in increasing order.) Then we apply this result to conforming elements,nonconforming elements and mixed elements of self-adjoint elliptic differential operators eigenvalue problems,and prove that the k-th approximate eigenvalue obtained by these methods converges to the k-th exact eigenvalue.     

(H,Ω)共轭函数理论

在非线性凸规划中,凸共轭函数理论对建立对偶理论起着重要作用.本文试图对这一凸共轭函数概念加以推广,建立一类广义的共轭函数理论-(H,(?))共轭函数理论.在凸分析中,一个函数的凸共轭是通过一簇线性函数确定的.事实上,设     

H~p(Ω)上的复合算子

本文首先给出H~p(Ω)上复合算子的特征和可逆的充要条件;然后给出范数估计式和C成为保距算子的条件;最后,在Hilbert空间H~2(Ω)中,给出保距算子的Wold分解。     

Direct and inverse spectral theory of singular left-definite Sturm–Liouville operators

We discuss direct and inverse spectral theory of self-adjoint Sturm–Liouville relations with separate boundary conditions in the left-definite setting. In particular, we develop singular Weyl–Titchmarsh theory for these relations. Consequently, we apply de Branges? subspace ordering theorem to obtain inverse uniqueness results for the associated spectral measure. The results can be applied to solve the inverse spectral problem associated with the Camassa–Holm equation.     

范畴M_R~l(Ω)若干性质的研究

研究了Ω-左R-模范畴中的余积及余等值子的性质,揭示了范畴M_R~l(Ω)与范畴M_R~l中余积之间的关系,刻画了范畴M_R~l(Ω)与范畴M_R~l中余等值子之间的关系,同时证明了范畴M_R~l(Ω)的余完备性.     

Normal Subgroups of B u Aut(Ω)

Aut(Ω) denotes the group of all order preserving permutations of the totally ordered set Ω, and if e ≤ u ∈ Aut(Ω), then B _u Aut(Ω) denotes the subgroup of all those permutations bounded pointwise by a power of u. It is known that if Aut(Ω) is highly transitive, then Aut(Ω) has just five normal subgroups. We show that if Aut(Ω) is highly transitive and u has just one interval of support, then B _u Aut(Ω) has normal subgroups, and there is a certain ideal of the lattice of subsets of (), the power set of the integers, such that the lattice of normal subgroups of every such Aut(Ω) is isomorphic to . To Bernhard Banaschewski on the occasion of his 80th birthday.