Substochastic semigroups for transport equations with conservative boundary conditions

We consider the free streaming operator associated with conservative boundary conditions. It is known that this operator (with its usual domain) admits an extension A which generates a C₀-semigroup in L¹. With techniques borrowed from the additive perturbation theory of substochastic semigroups, we describe precisely its domain and provide necessary and sufficient conditions ensuring to be stochastic. We apply these results to examples from kinetic theory.     

On the Essential Commutant of Toeplitz Operators in Several Complex Variables

Using the joint local mean oscillation, Jingbo Xia [13] showed that the essential commutant of , where is the subalgebra of L ^∞ generated by all functions which are bounded and have at most one discontinuity, is (QC). Even though Xia’s method cannot be used, we are able to generalize his result to Toeplitz operators in higher dimensions with a different approach. This result is stronger than the well-known result stating that the essential commutant of the full Toeplitz algebra is (QC).      

Anti-selfdual Lagrangians II: unbounded non self-adjoint operators and evolution equations

New variational principles based on the concept of anti-selfdual (ASD) Lagrangians were recently introduced in “AIHP-Analyse non linéaire, 2006”. We continue here the program of using such Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary boundary value problems of the form as well ass dissipative initial value evolutions of the form where is a convex potential on an infinite dimensional space, A is a linear operator and is any scalar. The framework developed in the above mentioned paper reformulates these problems as and respectively, where is an “ASD” vector field derived from a suitable Lagrangian L. In this paper, we extend the domain of application of this approach by establishing existence and regularity results under much less restrictive boundedness conditions on the anti-selfdual Lagrangian L so as to cover equations involving unbounded operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by transport operators with or without a diffusion term. Nassif Ghoussoub research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The author gratefully acknowledges the hospitality and support of the Centre de Recherches Mathématiques in Montréal where this work was initiated. Leo Tzou’s research was partially supported by a doctoral postgraduate scholarship from the Natural Science and Engineering Research Council of Canada.     

Schatten-von Neumann Hankel Operators on the Bergman Space of Planar Domains

In this paper we study the problem of the joint membership of Hφ and in the Schatten-von Neumann p-class when φ ∈ L∞(Ω) and Ω is a planar domain. We use a result of K. Zhu and the localization near the boundary to solve the problem. Finally, we recover a result of Arazy, Fisher and Peetre on the case with φ holomorphic.      

Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

On the continuity of the spectrum in certain Banach algebras

In this paper we describe some classes of linear operatorsTL(H) (mainly Toeplitz, Wiener-Hopf and singular integral) on a Hilbert spacesH such that the spectrum (T, L(H)) is continuous at the pointsT from these classes. We also describe some subalgebras of the algebras for which the spectrum (x,) becomes continuous at the pointsx when (x,) is restricted to the subalgebra . In particular, we show that the spectrum (x,) is continuous in Banach algebras with polynomial identities. Examples of such algebras are given.This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Boundary Value Problems for the Helmholtz Equation in an Octant

We consider a class of boundary value problems for the three-dimensional Helmholtz equation that appears in diffraction theory. On the three faces of the octant, which are quadrants, we admit first order boundary conditions with constant coefficients, linear combinations of Dirichlet, Neumann, impedance and/or oblique derivative type. A new variety of surface potentials yields 3 × 3 boundary pseudodifferential operators on the quarterplane that are equivalent to the operators associated to the boundary value problems in a Sobolev space setting. These operators are analyzed and inverted in particular cases, which gives us the analytical solution of a number of well-posed problems. Dedicated to Vladimir G. Maz’ya on the occasion of his 70th birthday     

Bounded Toeplitz Products on the Bergman Space of the Unit Ball in \mathbb{C}^n

We investigate necessary and sufficient conditions for boundedness of the operator on the Bergman space of the unit ball for n ≥ 1, where T_f is the Toeplitz operator. Those conditions are related to boundedness of the Berezin transform of symbols f and g. We construct the inner product formula which plays a crucial role in proving the sufficiency of the conditions.     

On the Fučik point spectrum for Schrödinger operators on {\mathbb{R}}^{N}

We investigate the Fučik point spectrum of the Schr?dinger operator when the potential V_λ has a steep potential well for sufficiently large parameter λ > 0. It is allowed that S_λ has essential spectrum with finitely many eigenvalues below the infimum of . We construct the first nontrivial curve in the Fučik point spectrum by minimax methods and show some qualitative properties of the curve and the corresponding eigenfunctions. As applications we establish some results on existence of multiple solutions for nonlinear Schr?dinger equations with jumping nonlinearity.      

On invertibility of matrix wiener-hopf operator on discrete linearly ordered Abelian group

A connection between an invertibility of a matrix Wiener-Hopf operator on a discrete linearly ordered Abelian group and a canonical factoribility of the matrix symbol of the operator is studied. A method of the paper [1] is extended to the case of the group . Necessary and sufficient conditions for a normal solvability, a generalized invertibility, and an invertibility of the operator with a strictly nonsingular 2×2 matrix symbol of a special kind are found. We also give necessary conditions of the factoribility and necessary and sufficient conditions of the canonical factoribility of this matrix symbol.     

Compact and finite rank operators satisfying a Hankel type equationT 2X=XT 1 *

In 1997, V. Pták defined the notion of generalized Hankel operator as follows: Given two contractions and , an operatorX: is said to be a generalized Hankel operator ifT ₂ X=XT ₁ ^* andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT ₁ andT ₂. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Pták's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.Dedicated to the memory of our master and friend Vlastimil Pták     

Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

Let A be a self-adjoint operator on a Hilbert space . Assume that the spectrum of A consists of two disjoint components σ₀ and σ₁. Let V be a bounded operator on , off-diagonal and J-self-adjoint with respect to the orthogonal decomposition where and are the spectral subspaces of A associated with the spectral sets σ₀ and σ₁, respectively. We find (optimal) conditions on V guaranteeing that the perturbed operator L = A + V is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on the variation of the spectral subspaces of A under the perturbation V. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a -symmetric perturbation is discussed. This work was supported by the Deutsche Forschungsgemeinschaft (DFG), the Heisenberg-Landau Program, and the Russian Foundation for Basic Research.     

Boundary value problems for some nonlinear systems with singular \phi-laplacian

Systems of differential equations of the form

Blow-up sets for linear diffusion equations in one dimension

We consider the heat equation in the half-line with Dirichlet boundary data which blow up in finite time. Though the blow-up set may be any interval [0,a], depending on the Dirichlet data, we prove that the effective blow-up set, that is, the set of points where the solution behaves like u(0,t), consists always only of the origin. As an application of our results we consider a system of two heat equations with a nontrivial nonlinear flux coupling at the boundary. We show that by prescribing the non-linearities the two components may have different blow-up sets. However, the effective blow-up sets do not depend on the coupling and coincide with the origin for both components.     

The Blow-up Rate of Solutions to Boundary Blow-up Problems for the Complex Monge–Ampère Operator

A regularity result for solutions to boundary blow-up problems for the complex Monge–Ampère operator in balls in is proved. For certain boundary blow-up problems on bounded, strongly pseudoconvex domains in with smooth boundary an estimate of the blow-up rate of solutions are given in terms of the distance to the boundary and the product of the eigenvalues of the Levi form.     

Noncoercive convection–diffusion elliptic problems with Neumann boundary conditions

We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation

Formulas for the Inverses of Toeplitz Matrices with Polynomially Singular Symbols

We consider large finite Toeplitz matrices with symbols of the form (1– cos )^p f() where p is a natural number and f is a sufficiently smooth positive function. By employing techniques based on the use of predictor polynomials, we derive exact and asymptotic formulas for the entries of the inverses of these matrices. We show in particular that asymptotically the inverse matrix mimics the Green kernel of a boundary value problem for the differential operator Submitted: June 20, 2003     

Gradient estimates for the Perona–Malik equation

We consider the Cauchy problem for the Perona–Malik equation

Minimal number of idempotent generators for certain algebras

It is known [KRS] that for each finitely generated Banach algebra there exists a numberN such that for eachn>N the matrix algebras can be generated by three idempotents. In this paper we show that the same statement is true for direct sums and , where is a finitely generated free algebra, i.e. polynomials in several non-commuting variables. These results are new even for algebras because the numberN we obtain here improves known estimates (see for example [R]). We show that the algebra can be generated by two idempotents if and only ifn _j =2 for eachj and is singly generated. Also we give an example of a free singly generated algebra for which can not be generated by two idempotents. But% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacuWFSeIqgaacaaaa!409A!\[{\tilde {\cal B}}\] can be generated by three idempotents for each singly generated free algebra .     

Ergodic Properties for Regular A-Contractions

The current article pleads for the possibility to obtain an orthogonal decomposition of a Hilbert space which is induced by a regular A-contraction defined in [9, 10], A being a positive operator on . The decomposition generalizes the well-known decomposition related to a contraction T of , which gives the ergodic character of T. This decomposition is being used to prove certain versions for regular A-contractions of the mean ergodic theorem, as well as a version of Patil’s theorem from [8]. Also, we characterize the solutions of corresponding functional equations in the range of A^1/2, by analogy with the result of Lin-Sine in [7].