Spectral approximation by L 2 discretization of the Laplace operator on manifolds of bounded geometry

The author considers a discretization of the p-form Laplacian on open complete Riemannian manifolds of bounded geometry. Following Dodziuk and Patodi [8], the eigenvalues below the essential spectrum together with their eigenforms are approximated by eigenvalues and eigencochains of a semicombinatorical Laplacian acting on L ₂-cochains. We obtain a similar result for a ray which is contained in the essential spectrum. An example of a manifold of bounded geometry which admits eigenvalues below the essential spectrum is constructed.     

On a conjecture of Hubner

In this paper we show that for a bounded linear operatorA on a complex Hilbert spaceH, the points on the boundary of the numerical range ofA with infinite curvature and unique tangent are in the essential spectrum ofA, thus positively answering a conjecture raised by Hubner in [3].     

On the peripheral spectrum of bounded positive semigroups on atomic Banach lattices

Generalizing a recent result of E.B. Davies [4], we show that generators of bounded positive C₀-semigroups on atomic Banach lattices with order continuous norm have trivial peripheral point spectrum. Moreover, we give examples that the peripheral spectrum can be any closed cyclic subset of . Received: 20 September 2005; revised: 23 January 2006     

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

A system of ordinary differential equations of mixed order on an interval (0, r₀) is considered, where some coefficients are singular at 0. Special cases have been dealt with by Kako , where the essential spectrum of an operator associated with a linearized MHD model was calculated, and more recently by Hardt , Mennicken and Naboko . In both papers this operator is a selfadjoint extension of an operator on sufficiently smooth functions. The approach in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly defined as the closure of an operator defined on sufficiently smooth functions. This closed operator can be written as a sum of a selfadjoint operator and a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of ℂ, and formulas for the calculation of the essential spectrum in terms of the coefficients are given.     

Collapsing and Dirac-Type Operators

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-manifold X/ with X = X//G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.     

Approximation of approximation numbers by truncation

LetA be a bounded linear operator onsome infinite-dimensional separable Hilbert spaceH and letA _n be the orthogonal compression ofA to the span of the firstn elements of an orthonormal basis ofH. We show that, for eachk1, the approximation numberss _k(A_n) converge to the corresponding approximation numbers _k(A) asn. This observation implies almost at once some well known results on the spectral approximation of bounded selfadjoint operators. For example, it allows us to identify the limits of all upper and lower eigenvalues ofA _n in the case whereA is selfadjoint. These limits give us all points of the spectrum of a selfadjoint operator which lie outside the convex hull of the essential spectrum. Moreover, it follows that the spectrum of a selfadjoint operatorA with a connected essential spectrum can be completely recovered from the eigenvalues ofA _n asn goes to infinity.     

Birkhoff Centre of a Semi Group

The concept of the Birkhoff centre of a semi group with 0 and 1 was introduced by U.M. Swamy and G.S. Murti [3], analogous to that of a bounded Poset [1], and proved that it is a Boolean algebra. This concept was extended to a semi group with sufficiently many commuting idempotents in [2] by G.S. Murti and proved that it is a relatively complemented distributive lattice. In this paper, we extend the above concept for a general semi group S and prove that the Birkhoff centre of any semi group S is a relatively complemented distributive lattice.AMS Subject Classification (1991): 06A12, 20M10.     

Multiplication Operators on the Lipschitz Space of a Tree

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space L{\mathcal{L}} of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L{\mathcal{L}} that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L{\mathcal{L}} are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of L{\mathcal{L}} we call the little Lipschitz space.     

On the non-round points of the boundary of the numerical range ∗

Let A be a bounded linear operator acting on a Hilbert space. It is well known (Donoghue, 1957) that comer points of the numerical range W(A) are eigenvalues of A. Recently (1995), this result was generalized by Hiibner who showed that points of infinite curvature on the boundary of W(A) lie in the spectrum of A. Hübner also conjectured that all such points are either corner points or lie in the essential spectrum of A. In this paper, we give a short proof of this conjecture.     

Validity of nonlinear geometric optics with times growing logarithmically

The profiles (a.k.a. amplitudes) which enter in the approximate solutions of nonlinear geometric optics satisfy equations, sometimes called the slowly varying amplitude equations, which are simpler than the original hyperbolic systems. When the underlying problem is conservative one often finds that the amplitudes are defined for all time and are uniformly bounded. The approximations of nonlinear geometric optics typically have percentage error which tends to zero uniformly on bounded time intervals as the wavelength tends to zero. Under suitable hypotheses when the amplitude is uniformly bounded in space and time we show that the percentage error tends to zero uniformly on time intervals which grow logarithmically. The proof relies in an essential way on the fact that one has a corrector to the leading term of geometric optics.

     

On some operators inc 0

Theorem. Let S be a bounded Suslin set in the plane. Then there is a bounded linear operator T in c_o, whose point spectrum σ _e (T)=S.     

On Oscillation of Functions with Bounded Spectral Band

In this paper, we consider extremal oscillatory properties of functions with bounded spectrum, i.e., with bounded support (in the sense of distributions) of the Fourier transform. For such functions f, we give criteria of extendability of }f} from the real axis to a function F on the complex plane with derivatives F ^(m) having no real zeros and without enlarging the width of spectrum. In particular, we give examples of functions $f$ from the real Paley–Wiener space such that every function f ^(m), m=0, 1,..., has a finite number of real zeros.     

On the conditions to control curvature tensors of Ricci flow

An important problem in the study of Ricci flow is to find the weakest conditions that provide control of the norm of the full Riemannian curvature tensor. In this article, supposing (M ⁿ , g(t)) is a solution to the Ricci flow on a Riemmannian manifold on time interval [0, T), we show that L^\fracn+22{L^\frac{n+2}{2}} norm bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor if M is closed and T < ∞. Next we prove, without condition T < ∞, that C ⁰ bound of scalar curvature and Weyl tensor can control the norm of the full Riemannian curvature tensor on complete manifolds. Finally, we show that to the Ricci flow on a complete non-compact Riemannian manifold with bounded curvature at t = 0 and with the uniformly bounded Ricci curvature tensor on M ⁿ  × [0, T), the curvature tensor stays uniformly bounded on M ⁿ  × [0, T). Hence we can extend the Ricci flow up to the time T. Some other results are also presented.     

On the invertibility and bounded extension of C 0 -semigroups

A well-known necessary and sufficient condition for the operator A to be the infinitesimal generator of a strongly continuous (C ₀ -) group is that both A and -A generate a C ₀ -semigroup. This seems to imply that one has to check the conditions in the Hille-Yosida Theorem for both A and -A . In this paper we show that this is not necessary. Given that A generates a C ₀ -semigroup we prove that a (weak) growth bound on the resolvent on a left half plane is sufficient to guarantee that A generates a group. This extends the recent result found by Liu, see [6]. Furthermore, we study when a generator of a bounded C ₀ -semigroup is the generator of a bounded group. The condition that we obtain is the same as found by Van Casteren in [2, 3], but we present a direct proof.     

Absolutely continuous spectrum of Schrödinger operators with potentials slowly decaying inside a cone

For a large class of multi-dimensional Schrödinger operators it is shown that the absolutely continuous spectrum is essentially supported by [0,∞). We require slow decay and mildly oscillatory behavior of the potential in a cone and can allow for arbitrary non-negative bounded potential outside the cone. In particular, we do not require the existence of wave operators. The result and method of proof extends previous work by Laptev, Naboko and Safronov.     

Fejér type theorems for Fourier-Stieltjes series

A theorem of Fejér states that if a periodic function F is of bounded variation on the closed interval [0, 2], then the nth partial sum of its formally differentiated Fourier series divided by n converges to ^-1 [F(x+0) - F(x-0)] at each point x. The generalization of this theorem for Fourier-Stieltjes series of nonperiodic functions of bounded variation is also known. These theorems can be interpreted in such a way that the terms of the Fourier-Stieltjes (or Fourier) series of F determine the atoms of the finite Borel measure on the torus T:= [0, 2) induced by an appropriate extension of F (or by F itself in the periodic case). The aim of the present paper is to extend all of these results to the Cesàro as well as Abel-Poisson means of Fourier-Stieltjes (or Fourier) series of a nonperiodic (or periodic) function F of bounded variation. At the end, we sketch a possible extension of these results to linear means defined by more general kernels.     

Invariance theory for infinite dimensional linear control systems

In this paper we characterize the situation wherein a subspaceS of a separable Hilbert state space is holdable under the abstract linear autonomous control system , whereA is the infinitesimal generator of aC ₀-semigroup of operators and whereB is a bounded linear operator mapping a Hilbert space intoX. WhenS D(A^*) is dense inS , it is shown that a necessary (but insufficient) condition for holdability is (1): . A stronger condition than (1) is shown to be sufficient for a type of approximate holdability. In the finite dimensional setting, (1) reduces to (A, B)-invariance, which is known to be equivalent to the existence of a (bounded) linear feedback control law which achieves holdability inS. We prove that this equivalence holds in infinite dimensions as well, whenA is bounded and the linear spacesS, B andS+ B are closed.In the unbounded case, our results are illustrated by the shift semigroup and by the heat equation on an infinite rod with distributed controls. In the bounded case, our example is an integro-differential control system.Research sponsored by the National Research Council of Canada under Grant A7271.Research sponsored by the National Research Council of Canada under Grant A4641.     

Reflexivity for Subnormal Systems With Dominating Spectrum in Product Domains

The question whether every subnormal tuple on a complex Hilbert space is reflexive is one of the major open problems in multivariable invariant subspace theory. Positive answers have been given for subnormal tuples with rich spectrum in the unit polydisc or the unit ball. The ball case has been extended by Didas [6] to strictly pseudoconvex domains. In the present note we extend the polydisc case by showing that every subnormal tuple with pure components and rich Taylor spectrum in a bounded polydomain is reflexive.     

On a characterization of the essential spectra of some matrix operators and application to two-group transport operators

In this paper, we investigate the essential approximate point spectrum and the essential defect spectrum of a 2 × 2 block operator matrix on a Banach space. Furthermore, we apply the obtained results to two-group transport operators in the Banach space L _p ([−a, a] × [−1, 1]) × L _p ([−a, a] × [−1, 1]), a > 0, p ≥ 1.