The Spectral Theory of Upper Triangular Matrices with Entries in a Banach Algebra

Assume that T is an upper triangular square matrix with entries in a unital Banach algebra. The main question studied here is: Under what conditions on the entries in T is it true that the spectrum of T is the union of the spectra of the diagonal entries of T? Also some results are proved concerning the Fredholm theroy of matrices with operator entries.     

A fixed point technique to refine a simple approximate eigenvalue and a corresponding eigenvector

Let T be a bounded operator on a Banach space X. Let λ₀ be a nonzero simple eigenvalue of a ‘nearby’ operator T₀ and let ?₀ be a corresponding eigenvector. Several modified versions of a fixed point scheme are given for iteratively refining the initial approximations λ₀ and ?₀ of an eigenvalue λ of T and a corresponding eigenvector ? Convergence of these schemes is proved by considering error bounds for the iterates. These bounds hold if a compact operator T is approximated in the norm or in a Collectively compact manner by a sequence (T₀) of bounded operators, and λ₀ and ?₀ are eigenelements of T_n0 for a fixed _n0 of ‘moderate’ size. Numerical examples are no included to illustrate the performation of various iteration schemes.     

On stability of index of Fredholm complexes on the C* -algebra

This article deals with the index of Fredholm complexes of Λ-operators of the Hilbert Λ-modulus on C*-algebra. For this class of operators necessary and sufficient conditions in order to be a Fredholm, are obtained. Based on these results, a notion of Fredholm complex and its index is introduced. For this index, a stability theorem related to various perturbations is proved. In the second part of the article, a completation of a semigroup Fredholm complexes is analysed. It is proved that the group K _G (X, Λ) is the completation of G ? Λ-fibration of the above group on the compact space X.     

Riemann summability of multi-dimensional trigonometric-fourier series

The d-dimensional classical Hardy spaces H_p(T ^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T ^d) to L_p(T ²) (d/(d+1)<p≤∞) and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. The same is proved for the conjugate Riemann sums. As a consequence we obtain that every function f∈L₁(T ^d) is a. e. Riemann summable to f, provided again that the limit is taken over a positive cone. This research was partly supported by the Hungarian Scientific Research Funds (OTKA) No F019633.     

Strong Solvability of a Unilateral Boundary Value Problem for Nonlinear Parabolic Operators

Strong solvability in Sobolev spaces is proved for a unilateral boundary value problem for nonlinear parabolic operators. The operator is assumed to be of Carathéodory type and to satisfy a suitable ellipticity condition; only measurability with respect to the independent variable X is required. The main tools of the proof are an estimate for the second derivatives of functions which satisfy the unilateral boundary conditions and the monotonicity of the operator − u _t with respect to Δu for the same functions.     

Some spectral properties of the streaming operator with general boundary conditions

This paper deals with the spectral study of the streaming operator with general boundary conditions defined by means of a boundary operator K. We study the positivity and the irreducibility of the generated semigroup proved in [M. Boulanouar, L’opérateur d’Advection: existence d’un C ₀-semi-groupe (I), Transp. Theory Stat. Phys. 31, 2002, 153–167], in the case ‖K‖ ⩾ 1. We also give some spectral properties of the streaming operator and we characterize the type of the generated semigroup in terms of the solution of a characteristic equation.     

On a Field-Theoretic Invariant for Extensions of Commutative Rings

Abstract

If R ? T is an extension of (commutative integral) domains, Λ(T/R) is defined as the supremum of the lengths of chains of intermediate fields in the extension k _R (Q ∩ R) ? k _T (Q), where Q runs over the prime ideals of T. The invariant Λ(T/R) is determined in case R and T are adjacent rings and in case Spec(R) = Spec(T) as sets. It is proved that if R is a domain with integral closure R′, then Λ(T/R) = 0 for all overrings T of R if and only if R′ is a Prüfer domain such that Λ(R′/R) = 0. If R ? T are domains such that the canonical map Spec(T) → Spec(R) is a homeomorphism (in the Zariski topology), then Λ(T/R) is bounded above by the supremum of the lengths of chains of rings intermediate between R and T. Examples are given to illustrate the sharpness of the results.     

On a transport operator arising in growing cell populations II. Cauchy problem

This paper deals with Rotenberg's models of cell populations with general boundary conditions. It is shown, first, that the associated Cauchy problem is governed by a C₀‐semigroup. Second, we have proved that if the boundary operator is positive, the transport semigroup is irreducible. And finally, a spectral decomposition of the solution into an asymptotic term was derived. Copyright © 2004 John Wiley & Sons, Ltd.     

Algebraic Topological Closure Operators

For any closure operator c there is a T_o-closure operator whose lattice of closed subsets are isomorphic to that of c. A correspondence between algebraic topological (T_o) closure operators on a nonempty set X and pre-orderes (partial orders) on X is established. Equivalent conditions are obtained for a T_o-lattice to be a complete atomic Boolean algebra and for the lattice of closed subsets of an algebraic topological closure operator to be a complete atomic Boolean algebra. Further it is proved that a complete lattice is an algebraic T_o-lattice if and only if it is isomorphic to the lattice of closed subsets of some algebraic topological closure operator on a suitable set.AMS Subject Classification (1991): 06A23, 54D65.     

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S?M?Xreal analytic submonifolds with codium ^R _MS=1,and let ω be a connected component of M\S. Let p∈S X_MT_M ^*X where T^* _Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *.

Under the assumption dim^R(T_pT_M ^*X? ν(p)) = 1, a theorem of vanishing for the cohomology groups H^jμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M.

Under the hypothesis dim^R(T_pT_S ^*X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.     

A link between sets of tensors stable under lamination and quasiconvexity

A link is found between quasiconvexity and the conditions for a set L of conductivity or elasticity tensors to be stable under lamination. These conditions, derived in the companion paper, are shown here to be equivalent to the condition that for every point B on the boundary of the set L an operator T_B dependent on the tangent plane and curvature of the set at B is a quasiconvex translation operator. A separate class of quasiconvex translation operators is obtained which are candidates for proving that L is stable under homogenization. The region stable under homogenization associated with any one of these operators shares a common boundary point and tangent plane with the set L and has curvature at that point not greater than the curvature of L. The conditions under which there exists a representative subclass of these operators such that the associated regions stable under homogenization wrap around L remains unresolved. It is proved that L can be characterized by minimizations of sums and dual energies in much the same way that convex sets can be characterized by their Legendre transforms. © 1994 John Wiley & Sons, Inc.     

An exact estimate of the boundary behavior of functions from hardy-sobolev classes in the critical case

In the critical case αp=n functions from the Hardy-Sobolev spacesH _α ^p (B ⁿ ) have a limit almost everywhere on the boundary along certain regions of exponential contact with the boundary. It is proved in the paper that the maximal operator associated with these regions is bounded as an operator fromH _α ^p (B ⁿ ) toL ^p (ϖB ⁿ ). Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 527–539, October, 1997. Translated by N. K. Kulman     

On a resolvent estimate of the Stokes equation with Neumann–Dirichlet‐type boundary condition on an infinite layer

This paper is concerned with the standard Lp estimate of solutions to the resolvent problem for the Stokes operator on an infinite layer with ‘Neumann–Dirichlet‐type’ boundary condition. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic and numerical study of electron flow spin polarization in 2D waveguides of variable cross‐section in the presence of magnetic field

We consider an infinite two‐dimensional waveguide that, far from the coordinate origin, coincides with a strip. The waveguide has two narrows of diameter ?. The narrows play the role of effective potential barriers for the longitudinal electron motion. The part of the waveguide between the narrows becomes a ‘resonator’, and there can arise conditions for electron resonant tunneling. A magnetic field in the resonator can change the basic characteristics of this phenomenon. In the presence of a magnetic field, the tunneling phenomenon is feasible for producing spin‐polarized electron flows consisting of electrons with spins of the same direction. We assume that the whole domain occupied by a magnetic field is in the resonator. An electron wave function satisfies the Pauli equation in the waveguide and vanishes at its boundary. Taking ? as a small parameter, we derive asymptotics for the probability T(E) of an electron with energy E to pass through the resonator, for the ‘resonant energy’ E_res, where T(E) takes its maximal value and for some other resonant tunneling characteristics. The asymptotic formulas contain some unknown constants. We find them by solving several auxiliary boundary value problems (independent of ?) in unbounded domains. Having the asymptotics with calculated constants, we can take it as numerical approximation to the resonant tunneling characteristics. Independently, we compute numerically the scattering matrix and compare the asymptotic and numerical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Periodic solutions of the stefan problem with hysteresis-type boundary conditions

We consider the Stefan problem with Dirichlet boundary conditions depending on a hysteresis functional where the free boundary is involved. We show existence of a positive valueT and existence of aT-periodic solution of the problem, provided the Stefan number is sufficiently small and the hysteresis functional is described by the elementary rectangular hysteresis loop. If in addition the Preisach hysteresis operator is Lipschitz-continuous we prove that every periodic solution must be stationary. Dedicated to Professor Avner Friedman on occasion of his 60th birthday supported by Humboldt Foundation Scholarship     

The generalized dock problem

We show existence and uniqueness for a linearized water wave problem in a two dimensional domain G with corner, formed by two semi-axes Γ₁ and Γ₂ which intersect under an angle α?∈?(0,?π]. The existence and uniqueness of the solution is proved by considering an auxiliary mixed problem with Dirichlet and Neumann boundary conditions. The latter guarantees the existence of the Dirichlet to Neumann map. The water wave boundary value problem is then shown to be equivalent to an equation like v_tt ?+?gΛv?=?P_t with initial conditions, where t stands for time, g is the gravitational constant, P means pressure and Λ is the Dirichlet to Neumann map. We then prove that Λ is a positive self-adjoint operator.     

Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems

The general problem studied has as a prototype the full non-linear Navier-Stokes equations for a slightly viscous compressible fluid including the heat transfer. The boundaries are of inflow-outflow type, i.e. non-characteristic, and the boundary conditions are the most general ones with any order of derivatives. It is assumed that the uniform Lopatinsky condition is satisfied. The goal is to prove uniform existence and boundedness of solution as the viscosity tends to zero and to justify the boundary layer asymptotics. The paper consists of two parts. In Part I the linear problem is studied. Here, uniform lower and higher order tangential estimates are derived and the existence of a solution is proved. The higher order estimates depend on the smoothness of coefficients; however this smoothness does not exceed the smoothness of the solution. In Part II the quasilinear problem is studied. It is assumed that for zero viscosity the overall initial-boundary value problem has a smooth solutionu ₀ in a time interval 0≦t≦T ₀. As a result the boundary laye, is weak and is uniformlyC ¹ bounded. This makes the linear theory applicable. an iteration scheme is set and proved to converge to the viscous solution. The convergence takes place for small viscosity and over the original time interval 0≦t≦T ₀.     

Some Applications of the Regularity and Irreducibility on Transport Theory

This paper is concerned with the spectral analysis of transport operator with general boundary conditions in L ₁-setting. This problem will be investigated under results from the theory of positive linear operators, irreducibility and regularity of the collision operator. The basic problems treated here are notions of essential spectra, spectral bound and leading eigenvalues.     

Uniqueness theorems for a nonlinear Tricomi problem and the related evolution problem

We consider an initial-boundary value problem for the non-linear evolution equation in a cylinder Q_t = Ω × (0, t), where T[u] = yu_xx + u_yy is the Tricomi operator and l(u) a special differential operator of first order. In [10] we proved the existence of a generalized solution of problem (1) and the existence of a generalized solution of the corresponding stationary boundary value problem (non-linear Tricomi problem) In this paper we give sufficient conditions for the uniqueness of these solutions.