On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

We prove a Mihlin–type multiplier theorem for operator–valued multiplier functions on UMD–spaces. The essential assumption is R–boundedness of the multiplier function. As an application we give a characterization of maximal –regularity for the generator of an analytic semigroup in terms of the R–boundedness of the resolvent of A or the semigroup . Received July 19, 1999 / Revised July 13, 2000 / Published online February 5, 2001     

Product Semigroups and Extrapolation Spaces

A new formulation of the theory of extrapolation spaces is used to recast a theorem on the resolvent of the sum of a generator of a semigroup and a Hille–Yosida operator and to prove a regularity result related to an abstract equation.     

A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices

We present simple proofs of several basic facts of the global regime (the existence and the form of the nonrandom limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices whose probability law involves the Gaussian distribution. The main difference with previous proofs is the systematic use of the Poincare-Nash inequality, allowing us to obtain the O(n ⁻²) bounds for the variance of the normalized trace of the resolvent that are valid up to the real axis in the spectral parameter. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 790–817, June, 2005.     

Error growth analysis via stability regions for discretizations of initial value problems

This paper deals with numerical methods for the solution of linear initial value problems. Two main theorems are presented on the stability of these methods. Both theorems give conditions guaranteeing a mild error growth, for one-step methods characterized by a rational function ϕ(z). The conditions are related to the stability regionS={z:z∈ℂ with |ϕ(z)|≤1}, and can be viewed as variants to the resolvent condition occurring in the reputed Kreiss matrix theorem. Stability estimates are presented in terms of the number of time stepsn and the dimensions of the space. The first theorem gives a stability estimate which implies that errors in the numerical process cannot grow faster than linearly withs orn. It improves previous results in the literature where various restrictions were imposed onS and ϕ(z), including ϕ′(z)≠0 forz∈σS andS be bounded. The new theorem is not subject to any of these restrictions. The second theorem gives a sharper stability result under additional assumptions regarding the differential equation. This result implies that errors cannot grow faster thann ^β, with fixed β<1. The theory is illustrated in the numerical solution of an initial-boundary value problem for a partial differential equation, where the error growth is measured in the maximum norm.     

A note on Linnik’s theorem on quadratic non-residues

Archiv der Mathematik - We present a short and purely combinatorial proof of Linnik’s theorem: for any $$\varepsilon &gt;0$$ there exists a constant $$C_\varepsilon $$ such that for any...     

Resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds I: Resolvent construction at high energy

This is the first in a series of papers in which we investigate the resolvent and spectral measure on non-trapping asymptotically hyperbolic manifolds with applications to the restriction theorem, spectral multiplier results and Strichartz estimates. In this first paper, we construct the high energy resolvent on general non-trapping asymptotically hyperbolic manifolds, using semiclassical Lagrangian distributions and semiclassical intersecting Lagrangian distributions, together with the 0-calculus of Mazzeo-Melrose.

Our results generalize recent work of Melrose, Sá Barreto and Vasy, which applies to metrics close to the exact hyperbolic metric. We note that there is an independent work by Y. Wang which also constructs the high-energy resolvent.     

Spectral boundary value problems in Lipschitz domains for strongly elliptic systems in Banach spaces H p σ and B p σ

In a bounded Lipschitz domain, we consider a strongly elliptic second-order equation with spectral parameter without assuming that the principal part is Hermitian. For the Dirichlet and Neumann problems in a weak setting, we prove the optimal resolvent estimates in the spaces of Bessel potentials and the Besov spaces. We do not use surface potentials. In these spaces, we derive a representation of the resolvent as a ratio of entire analytic functions with sharp estimates of their growth and prove theorems on the completeness of the root functions and on the summability of Fourier series with respect to them by the Abel-Lidskii method. Preliminarily, such questions for abstract operators in Banach spaces are discussed. For the Steklov problem with spectral parameter in the boundary condition, we obtain similar results. We indicate applications of the resolvent estimates to parabolic problems in a Lipschitz cylinder. We also indicate generalizations to systems of equations. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 42, No. 4, pp. 2–23, 2008 Original Russian Text Copyright ? by M. S. Agranovich To dear Israel Moiseevich Gelfand in connection with his 95th birthday Supported by RFBR grant no. 07-01-00287.     

Borel complexity and computability of the Hahn–Banach Theorem

The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof of the theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach–Alaoglu Theorem we can show that computing a Hahn–Banach extension cannot be harder than finding a zero in a compact metric space. This allows us to conclude that the Hahn–Banach extension operator is -computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn–Banach Theorem for those functionals and subspaces which admit unique extensions. This work has been partially supported by the National Research Foundation (NRF) Grant FA2005033000027 on “Computable Analysis and Quantum Computing”. An extended abstract version has been published in the conference proceedings [7].     

Asymptotic behavior of the resolvent of an unstable volterra equation with kernel depending on the difference of the arguments

We present the structure of the resolvent of a difference kernel, which allows us to study the asymptotic behavior of the solution of the renewal equation for a given asymptotic behavior of the constant term. An asymptotic representation for the resolvent is obtained under minimal requirements on the moments of the kernel. Similar results are given for integro-differential equations. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 88–94, July, 1997. Translated by M. A. Shishkova     

Some remarks on Nagumo’s theorem

We provide a simpler proof for a recent generalization of Nagumo’s uniqueness theorem by A. Constantin: On Nagumo’s theorem. Proc. Japan Acad., Ser. A 86 (2010), 41–44, for the differential equation x′ = f(t, x), x(0) = 0 and we show that not only is the solution unique but the Picard successive approximations converge to the unique solution. The proof is based on an approach that was developed in Z. S. Athanassov: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jap. 35 (1990), 351–367. Some classical existence and uniqueness results for initial-value problems for ordinary differential equations are particular cases of our result.     

Optimal polynomial decay of functions and operator semigroups

We characterize the polynomial decay of orbits of Hilbert space C ₀-semigroups in resolvent terms. We also show that results of the same type for general Banach space semigroups and functions obtained recently in Batty and Duyckaerts (J Evol Eq 8:765–780, 2008) are sharp. This settles a conjecture posed in Batty and Duyckaerts (2008).     

The Abstract Titchmarsh-Weyl M-function for Adjoint Operator Pairs and its Relation to the Spectrum

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction A_B of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces and such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples – one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible. James Hinchcliffe and Serguei Naboko wish to thank the British EPSRC for financial support under grant EP/C008324/1 “Spectral Problems on Families of Domains and Operator M-functions”. Serguei Naboko wishes to thank the Russian RFBR for grant 06-01-00219. All authors wish to thank INTAS for financial support under INTAS Project No. 051000008-7883. The authors wish to thank the referee for many helpful comments.     

A Skolem–Mahler–Lech theorem in positive characteristic and finite automata

Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the Skolem–Mahler–Lech theorem. Lech gave a counterexample to a similar statement in positive characteristic. We will present some more pathological examples. We will state and prove a correct analog of the Skolem–Mahler–Lech theorem in positive characteristic. The zeroes of a recurrence sequence in positive characteristic can be described using finite automata.     

Generalized Browder’s Theorem and SVEP

A bounded operator a Banach space, is said to verify generalized Browder’s theorem if the set of all spectral points that do not belong to the B-Weyl’s spectrum coincides with the set of all poles of the resolvent of T, while T is said to verify generalized Weyl’s theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying generalized Browder’s theorem, or generalized Weyl’s theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H ₀(λI − T) as λ belongs to certain subsets of . In the last part we give a general framework for which generalized Weyl’s theorem follows for several classes of operators.     

General System of A-Monotone Nonlinear Variational Inclusion Problems with Applications

Based on the notion of A–monotonicity, the solvability of a system of nonlinear variational inclusions using the resolvent operator technique is presented. The results obtained are new and general in nature.     

Characteristic Conditions for the Generation of α-Times Resolvent Families on a Hilbert Space

In 2000,Shi and Feng gave the characteristic conditions for the generation of C0semigroups on a Hilbert space.In this paper,we will extend them to the generation of α-times resolvent operator families.Such characteristic conditions can be applied to show rank-1 perturbation theorem and relatively-bounded perturbation theorem for α-times resolvent operator families.     

Cartan decomposition of the moment map

We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.First author partially supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft and the DFG Schwerpunk program Globale Methoden in der komplexen Geometrie.Second author partially supported by NSA grant H98230–04–01–0070.     

Stability properties for solution operators

We study stability properties of certain evolution equations including the fractional Cauchy problem. Under some spectral assumptions these equations are governed either by a resolvent or a regularized resolvent or a k-convoluted semigroup. We investigate the long time behavior for bounded solutions by a direct application of the ergodic theorems for regularized resolvents of Lizama and Prado (J. Approx. Theory 122:42–61, 2003), Prado (Semigroup Forum 73:243–252, 2006). We apply our results to the qualitative study of the fractional diffusion-wave equation on L ^p (ℝ). The author is partially supported under FONDECYT Grant n^o 1070127.     

On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition

We consider a waveguide modeled by the Laplacian in a straight planar strip. The Dirichlet boundary condition is taken on the upper boundary, while on the lower boundary we impose periodically alternating Dirichlet and Neumann condition assuming the period of alternation to be small. We study the case when the homogenization gives the Neumann condition instead of the alternating ones. We establish the uniform resolvent convergence and the estimates for the rate of convergence. It is shown that the rate of the convergence can be improved by employing a special boundary corrector. Other results are the uniform resolvent convergence for the operator on the cell of periodicity obtained by the Floquet–Bloch decomposition, the two terms asymptotics for the band functions, and the complete asymptotic expansion for the bottom of the spectrum with an exponentially small error term.