On spectral variations under relatively bounded perturbations

Linear operators in Banach and Hilbert spaces are considered. Bounds for the spectrum are established under relatively bounded perturbations. An application to nonselfadjoint differential operators is discussed.     

On bounded time-dependent perturbations of operator cosine functions

In this note it is shown that the Cauchy problem associated with the infinitesimal generatorA of a strongly continuous operator cosine function remains uniformly well-posed under bounded time-dependent perturbations ofA.     

On direct limit classes of algebras

We investigate classes of algebras which can be obtained by a direct limit construction from an algebra. We generalize some results from monounary algebras. Supported by grant VEGA 2/5065/5.     

On spectral variations under bounded real matrix perturbations

Summary In this paper we investigate the set of eigenvalues of a perturbed matrix {ie509-1} whereA is given and ^{n × n}, ||< is arbitrary. We determine a lower bound for thisspectral value set which is exact for normal matricesA with well separated eigenvalues. We also investigate the behaviour of the spectral value set under similarity transformations. The results are then applied tostability radii which measure the distance of a matrixA from the set of matrices having at least one eigenvalue in a given closed instability domain _b.     

On convex perturbations with a bounded isotropic constant

Let be a convex body and ɛ > 0. We prove the existence of another convex body , whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than , where c > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain. The author is a Clay Research Fellow, and was also supported by NSF grant #DMS-0456590. Received: November 2005; Accepted: February 2006     

On some classes of bounded univalent mappings in several complex variables

Let B ⁿ be the Euclidean unit ball in C ⁿ . In this paper, we study several properties of strongly starlike mappings of order α (0 < α < 1) and bounded convex mappings on B ⁿ . We prove that K-quasiregular strongly starlike mappings of order α on B ⁿ have continuous and univalent extensions to ${\overline{B}^n}$ . We show that bounded convex mappings on B ⁿ are strongly starlike of some order α. We give a coefficient estimate for K-quasiregular strongly starlike mappings of order α on B ⁿ . Finally, we give examples of strongly starlike mappings of order α and bounded convex mappings on B ⁿ .     

On exponential stability of infinite dimensional linear systems with bounded or unbounded perturbations

In this paper, we study the question of stabilization of uncertain systems governed by evolution equations in Ililbert space. It is shown that exponential stability can be achieved by choice of suitable state feedback controls even in the presence of bounded or relatively bounded (uncertain) perturbations. Our proof is based on perturbation theory of semigroups. The results are illustrated by two examples involving heat equation and wave equation.     

Uniformly bounded limit of fractional homomorphisms

We show that a bounded homomorphism is equivalent to a uniformly bounded family of fractional homomorphisms for any 0$">. We add this characterization to the Widder-Arendt-Kisynski theorem and relate it to -times integrated semigroups.

     

On the Chern-Simons limit for a Maxwell-Chern-Simons model on bounded domains

This paper concerns the Chern-Simons limit for the Abelian Maxwell-Chern-Simons model on bounded domains with vanishing gauge fields. We prove that every sequence of solutions of the Maxwell-Chern-Simons equations has a subsequence converging to a solution of the Chern-Simons equation in any Ck norms. We also show that the Maxwell-Chern-Simons equations with the nontopological type boundary condition do not admit any nontrivial solutions on star-shaped domains.     

On the limit cycles available from polynomial perturbations of the Bogdanov-Takens Hamiltonian

The displacement map related to small polynomial perturbations of the planar Hamiltonian systemdH=0 is studied in the elliptic caseH=1/2y ²+1/2x ²−1/3x ³. An estimate of the number of isolated zeros for each of the successive Melnikov functionsM _k(h),k=1, 2,…is given in terms of the orderk and the maximal degreen of the perturbation. This sets up an upper bound to the number of limit cycles emerging from the periodic orbits of the Hamiltonian system under polynomial perturbations. Research partially supported by grant MM810/98 from the NSF of Bulgaria and MURST, Italy.     

Finite rank,relatively bounded perturbations of semigroups generators

Summary This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator A_F=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b H. While Part I studied the question of generation of a s.c. semigroup on H by A_F and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of A_F, given A and a H, via a suitable vector b H; alternatively, given A, via a suitable pair of vectors a, b H; (ii) spectrality of A_F—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {_n} of A, the given vector a H and a given suitable sequence {_n} of nonzero complex numbers, which guarantee the existence of a suitable vector b H such that A_F possesses the following two desirable properties: (i) the eigenvalues of A_F are precisely equal to _n+_n; (ii) the corresponding eigenvectors of A_F form a Riesz basis (a fortiori, A_F is spectral). While finitely many _ns can be preassigned arbitrarily, it must be however that _n 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.Research partially supported by Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

On the strict monotonicity of spectral radii for classes of bounded positive linear operators

Strict monotonicity of the spectral radii of bounded, positive, ordered linear operators is investigated. It is well-known that under reasonable assumptions, the spectral radii of two ordered positive operators enjoy a non-strict inequality. It is also well-known that a “strict” inequality between operators does not imply strict monotonicity of the spectral radii in general—some additional structure is required. We present a number of sufficient conditions on both the cone and the operators for such a strict ordering to hold which generalise known results in the literature, and have utility in comparison arguments, ubiquitous in positive systems theory.