Similarity classification of holomorphic curves

Let H be a complex separable Hilbert space. For Ω an open connected subset of C, we shall say that a map is a holomorphic curve, if there exist n holomorphic H-valued functions γ₁,γ₂,…,γn on Ω such that f(λ)=?{γ₁(λ),…,γn(λ)}, ∀λ∈Ω, where Gr(n,H) denotes the Grassmann manifold, the set of all n-dimensional subspaces of H.In the paper, we give a similarity classification of some holomorphic curves by using the K-group of its commutant algebra as an invariant.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

Joint similarity problems and the generation of operator algebras with bounded length

There is a pair of commuting operators (T ₁,T ₂) on Hilbert space such that eachT ₁ andT ₂ is similar to a contraction but the pair (T ₁,T ₂) is not similar to a pair of contractions. There is a pair of commuting unitarizable representations (₁,₂) on the free group withN2 generators such that (₁,₂) is not similar to a pair of unitary representations. In connection with these examples, we introduce and study a notion of length for aC ^*-algebra (or an operator algebra) generated by two subalgebras, which is analogous to the minimum length of a word in the generators of a group.Partially supported by the N.S.F.     

Defect operators, defect functions and defect indices for analytic submodules

This paper mainly concerns defect operators and defect functions of Hardy submodules, Bergman submodules over the unit ball, and Hardy submodules over the polydisk. The defect operator (function) carries key information about operator theory (function theory) and structure of analytic submodules. The problem when a submodule has finite defect is attacked for both Hardy submodules and Bergman submodules. Our interest will be in submodules generated by polynomials. The reason for choosing such submodules is to understand the interaction of operator theory, function theory and algebraic geometry.     

Small compact perturbation of strongly irreducible operators

An operatorT onH is called strongly irreducible ifT is not similar to any reducible operators. In this paper, we shall say yes to answer the following question raised by D. A. Herrero.Given an operatorT with connected spectrum (T) and a positive number , can we find a compact operatorK with K < such thatT+K is strongly irreducible?Supported by National Natural Science Foundation of China(19901011), Mathematical Center of State Education Commission of China and 973 Project of China     

Elementary operators and the Aluthge transform

We characterize essential normality for certain elementary operators acting on the Hilbert-Schmidt class. We find the Aluthge transform of an elementary operator of length one. We show that the Aluthge transform of an elementary 2-isometry need not be a 2-isometry. We also characterize hermitian elementary operators of length 2.     

Spectral measures,II: Characterization of scalar operators

We continue the development of part I. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (continuous or otherwise) to be spectral. A uniqueness problem is pointed out and the function calculus is extended to the case of several variables. A Radon—Nikodym theorem is proved.     

Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space

In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

A characterization of boundary conditions yielding maximal monotone operators

We provide a characterization for maximal monotone realizations for a certain class of (nonlinear) operators in terms of their corresponding boundary data spaces. The operators under consideration naturally arise in the study of evolutionary problems in mathematical physics. We apply our abstract characterization result to Port–Hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco-elasticity.     

Compact and inessential perturbations of operators with closed range

The size of the perturbation class {SL(E)S has closed range}+I(E) is studied, whereE is a Banach space andI(E) stands for various classical operator ideals. For instance, it is shown for the ideal consisting of the inessential operators that the resulting perturbation class does not exhaust the class of bounded linear operators under natural structural conditions onE. It is known from a recent result of Gowers and Maurey that some conditions are needed.Partially supported by the Academy of Finland     

Asymptotic behavior of resolvents of coaccretive operators in the Hilbert ball

We study the resolvents of coaccretive operators in the Hilbert ball, with special emphasis on the asymptotic behavior of their compositions and metric convex combinations. We consider the case where the given coaccretive operators share a common fixed point inside the ball, as well as the case where they share a common sink point on its boundary. We establish weak convergence in the former case and strong convergence in the latter. We also present two related convergence results for a continuous implicit scheme.     

Products of orthogonal projections as Carleman operators

In this paper, we consider the product of two orthogonal projectionsP andQ on a separable, infinite dimensional Hilbert spaceH. For the operatorQP, there holds the dichotomy:QP is either a Carleman operator or a semi-Fredholm operator with finite defect. Both cases are characterized in terms of the dimensions of the ranges and null spaces ofP andQ and some of their intersections. This extends the case, whereP andQ are the special projections onto the subspaces of time- and band-limited functions inL ²() resp., first considered by Slepian, Pollak and Landau.     

Perturbed unstable variational inequalities with unbounded operators on approximately given sets

The stability and convergence of the solutions of perturbed and regularized variational inequality to the solutions of the primary (unstable a priori) variational inequality with proper monotone operator are investigated. All the objects of inequality: the operatorA, the right-hand partf and the set of constrains are to be perturbed. At the same time no assumptions of boundedness and smoothness of the operatorA are used. The connection between the parameters of perturbations, which guarantees strong convergence of approximate solutions, is established. It is proved that the existence of the solution to the unperturbed variational inequality is necessary and sufficient condition for convergence of the regularized perturbed inequality solutions.This research was supported in part by the Ministry of Science Grant 3481-1-91 and by the Ministry of Absorption Center for Absorption in Science.     

A global bifurcation index for set-valued perturbations of Fredholm operators

In the paper the homotopy invariant detecting global bifurcations of solutions to multi-parameter equations involving general set-valued perturbations of linear Fredholm operators of non-negative index is introduced. Some applications to the existence problems for differential inclusions are provided.     

Sequences in the range of a vector measure and Banach spaces satisfying Π1(X,l 1) = Π2(X,l 1)

If X is a Banach space such that Π₁(X,l ₁) = Π₂ (X, l ₁), we prove the following results: 1) A bounded sequence (x _n ) lies inside the range of some X-valued measure if and only if the operator (α _n ) ? l ₁ → Σ _n α _n x _n ? X is 1-summing, and 2) If A is a bounded subset of X lying in the range of some X ^??-valued measure, then A is necessarily contained in the range of some X-valued measure.     

Generalized inverses over Banach algebras

LetA be a matrix over a complex commutative unital Banach algebra. We give necessary and sufficient conditions forA to have a generalized inverse. Moreover, if the Banach algebra has a symmetric involution, these are also necessary and sufficient conditions forA to admit the Moore-Penrose inverse.Partially supported by NSF Grant DMS-8802593     

COMPLEMENTED RANGE SPACES

1. Abstract

Most of the well known perturbation results hold for linear operators whose nullity or defect is finite, which implies the existence of topological complements to the null and range spaces respectively. This paper investigates properties associated with the stability of the complementability of the range space under various perturbations.