Spectralizable Operators

We introduce the notion of spectralizable operators. A closed operator A in a Hilbert space is called spectralizable if there exists a non-constant polynomial p such that the operator p(A) is a scalar spectral operator in the sense of Dunford. We show that such operators belongs to the class of generalized spectral operators and give some examples where spectralizable operators occur naturally. Vladimir Strauss gratefully acknowledges support by DFG, Grant No. TR 903/3-1.     

Unitary mappings between multiresolution analyses of L2 (R) and a parametrization of low-pass filters

This article provides classes of unitary operators of L²(R) contained in the commutant of the Shift operator, such that for any pair of multiresolution analyses of L²(R) there exists a unitary operator in one of these classes, which maps all the scaling functions of the first multiresolution analysis to scaling functions of the other. We use these unitary operators to provide an interesting class of scaling functions. We show that the Dai-Larson unitary parametrization of orthonormal wavelets is not suitable for the study of scaling functions. These operators give an interesting relation between low-pass filters corresponding to scaling functions, which is implemented by a special class of unitary operators acting on L²([−π, π)), which we characterize. Using this characterization we recapture Daubechies' orthonormal wavelets bypassing the spectral factorization process. Acknowledgements and Notes. Partially supported by NSF Grant DMS-9157512, and Linear Analysis and Probability Workshop, Texas A&M University Dedicated to the memory of Professor Emeritus Vassilis Metaxas.     

Periodic solutions of second order differential equations in Banach spaces

We use operator-valued Fourier multiplier theorems to study second order differential equations in Banach spaces. We establish maximal regularity results in L^p and C^s for strong solutions of a complete second order equation. In the second part, we study mild solutions for the second order problem. Two types of mild solutions are considered. When the operator A involved is the generator of a strongly continuous cosine function, we give characterizations in terms of Fourier multipliers and spectral properties of the cosine function. The results obtained are applied to elliptic partial differential operators. The first author is supported in part by Convenio de Cooperación Internacional (CONICYT) Grant # 7010675 and the second author is partially financed by FONDECYT Grant # 1010675     

Index theory on locally homogeneous spaces

We describe how the equivariant K homology class of an invariant elliptic operator on a homogeneous space of a linear semisimple Lie group determines the L ²-index of the associated operator on a finite volume locally homogeneous space. The machinery of equivariant K homology and of KK theory can be used to prove theorems about L ²-indices. We give an application motivated by the problem of calculating multiplicities of subrepresentations of quasi-regular representations.Supported by the National Science Foundation under Grant No. DMS-8903472.Supported by the National Science Foundation under Grant No. DMS-8901436.     

Feedback operator for a pseudoparabolic optimal control problem

The feedback operator of a linear pseudoparabolic problem with quadratic criterion is obtained by decoupling of the optimality condition. The feedback operator is shown to be related to the solution of a Riccati equation formulated in theB*-algebra of bounded linear operators onL ²(). This approach shows that the linear feedback operator may be considered as a bounded operator fromL ²() intoH ₀ ¹ (). Finally, we give a theorem establishing the convergence behavior for the feedback operators for these problems as they formally approach an analogous problem of parabolic type.This work was supported in part by the National Science Foundation, Grant No. MCS-7902037.     

On Symmetrizable Operators on Hilbert Spaces

This paper, motivated by transport theory, deals with spectral properties of operators G on a complex Hilbert space H such that SG is self-adjoint where S is a nonnegative operator: We give several lower bounds of the spectral radius of G and determine the latter in some cases. We derive the whole spectrum for power compact G by means of Lagrange multiplier theory. We find out spectral connections between G and SG. We give a (spectral) stability estimate for symmetrizable operators in terms of the spectral radius of the perturbation.     

Ergodic properties of operators in some semi-Hilbertian spaces

This article deals with linear operators T on a complex Hilbert space ?, which are bounded with respect to the seminorm induced by a positive operator A on ?. The A-adjoint and A ^1/2-adjoint of T are considered to obtain some ergodic conditions for T with respect to A. These operators are also employed to investigate the class of orthogonally mean ergodic operators as well as that of A-power bounded operators. Some classes of orthogonally mean ergodic or A-ergodic operators, which come from the theory of generalized Toeplitz operators are considered. In particular, we give an example of an A-ergodic operator (with an injective A) which is not Cesàro ergodic, such that T ^?* is not a quasiaffine transform of an orthogonally mean ergodic operator.     

Topological Structure of the Set of Weighted Composition Operators on Weighted Bergman Spaces of Infinite Order

We consider the topological space of all weighted composition operators on weighted Bergman spaces of infinite order endowed with the operator norm. We show that the set of compact weighted composition operators is path connected. Furthermore, we find conditions to ensure that two weighted composition operators are in the same path connected component if the difference of them is compact. Moreover, we compare the topologies induced by L(H^∞) and L(H^∞_v) on the space of bounded composition operators and give a sufficient condition for a composition operator to be isolated.     

A criterion for Hill operators to be spectral operators of scalar type

We derive necessary and sufficient conditions for a Hill operator (i.e., a one-dimensional periodic Schrö dinger operator) H = ?d ² /dx ² + V to be a spectral operator of scalar type. The conditions show the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. In the course of our analysis, we also establish a functional model for periodic Schrödinger operators that are spectral operators of scalar type and develop the corresponding eigenfunction expansion.The problem of deciding which Hill operators are spectral operators of scalar type appears to have been open for about 40 years.     

L p-Theory of elliptic differential operators on manifolds of bounded geometry

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L ^pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L ^p-norms of the operators of this semigroup. Some spectral properties of such operators in L ^pspaces are also studied.     

B-fredholm and spectral properties for multipliers in Banach algebras

The main purpose of this paper is to study spectral and B-Fredholm properties of a multiplierT acting on a semi-simple regular tauberian commutative Banach algebraA. We show thatT is a B-Fredholm operator if and only ifT is a semi B-Fredholm operator, and in this case we have the indexind(T)=0. Moreover we give some spectral properties for multipliers. Spectral mapping theorems for the Weyl’s and B-Weyl spectrum of a multiplier are also considered. Furthermore we show that Weyl’s theorem and generalized Weyl’s theorem hold for a multiplierT. Finally we give sufficient conditions for a multiplier to be a product of an invertible and an idempotent operators.     

Weighted Composition Operators from H ∞ to the Bloch Space of a Bounded Homogeneous Domain

Let D be a bounded homogeneous domain in ℂ ⁿ . In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space H ^∞(D) into the Bloch space of D. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if D is a bounded symmetric domain, the bounded multiplication operators from H ^∞(D) to the Bloch space of D are the operators whose symbol is bounded.     

Remarks on generators of analytic semigroups

This paper contains two new characterizations of generators of analytic semigroups of linear operators in a Banach space. These characterizations do not require use of complex numbers. One is used to give a new proof that strongly elliptic second order partial differential operators generate analytic semigroups inL ^p , 1<p<∞, while the sufficient condition in the other characterization is meaningful in the case of nonlinear operators. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under Grant No. MCS78-01245.     

On Multiple Singular Integrals along Polynomial Curves with Rough Kernels

This paper is devoted to the study of a class of singular integral operators defined by polynomial mappings on product domains. Some rather weak size conditions, which imply the Lp boundedness of these singular integral operators as well as the corresponding maximal truncated singular integral operators for some fixed 1〈p〈 ∞,are given.     

Commutants modulo the compact operators of certain CSL algebras II

The commutant modulo compacts, or essential commutant, of a reflexive algebra with commutative subspace lattice is a C^* algebra which is the sum of the compact operators in L(H) and a C^* subalgebra of the core. We give a characterization of the essential commutant of a separably acting CSL algebra in terms of properties of the spectral measure of an operator in the intersection of the essential commutant and the core. This is used to determine some sufficient conditions on the lattice for when the essential commutant is norm generated by the projections it contains.     

Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators

We consider nonself-adjoint nondissipative trace class additive perturbations L=A+iV of a bounded self-adjoint operator A in a Hilbert space ,H. The main goal is to study the properties of the singular spectral subspace N _i ⁰ of L corresponding to part of the real singular spectrum and playing a special role in spectral theory of nonself-adjoint nondissipative operators.To some extent, the properties of N _i ⁰ resemble those of the singular spectral subspace of a self-adjoint operator. Namely, we prove that L and the adjoint operator ,L ^* are weakly annihilated by some scalar bounded outer analytic functions if and only if both of them satisfy the condition N _i ⁰ =H. This is a generalization of the well-known Cayley identity to nonself-adjoint operators of the above-mentioned class.     

Integral expressions of Lyapunov exponents for autonomous ordinary differential systems

In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space ℝ ^d , not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces. This work was supported by the National Natural Science Foundation of China (Grant No. 10671088) and the Major State Basic Research Development Program of China (Grant No. 2006CB805903)     

Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type

We show that maximal operators formed by dilations of Mikhlin- Hörmander multipliers are typically not bounded on L^p(^d). We also give rather weak conditions in terms of the decay of such multipliers under which L^p boundedness of the maximal operators holds.Christ, Grafakos and Seeger were supported in part by NSF grants. Honzík was supported by 201/03/0931 Grant Agency of the Czech Republic     

Toeplitz operators hyponormal with the unilateral shift

For the unilateral shift operator U on the Hardy space H²(T), we describe conditions on operators T, acting on H²(T), that are necessary and sufficient for the pair (U, T) to be jointly hyponormal. One necessary condition is that T be a Toeplitz operator. Consequently, we study certain nonanalytic symbols that give rise to Toeplitz operators hyponormal with the shift, and thereby obtain examples of noncommuting, jointly hyponormal pairs.Supported in part by a research grant from NSERC     

A conjecture on convolution operators,and a non-Dunford-Pettis operator onL 1

There exists a non-Dunford-Pettis operator fromL ¹ into a Banach latticeE that does not contain a copy ofc ₀ orL ¹. This problem is related to regularisation properties of convolution operators onL ¹. Work partially supported by an N.S.F. Grant.