On closures of similarity orbits of essentially nilpotent operators of order two

Several results are obtained on the structure of the closure of the similarity orbit of a compact perturbation of the ampliation of a nilpotent Jordan cell, including a complete answer for the case of an algebraic essentially nilpotent operator of order two.The research of both authors was partially supported by Grants of the National Science Foundation.     

On weak orbits of operators

Let T be a completely nonunitary contraction on a Hilbert space H with r(T)=1. Let an>0, an→0. Then there exists x∈H with |〈Tnx,x〉|?an for all n. We construct a unitary operator without this property. This gives a negative answer to a problem of van Neerven.     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.     

On closures of joint similarity orbits

For an n-tuple T=(T₁,..., T_n) of operators on a Hilbert spacexxHx, the joint similarity orbit of T isxxSx(T)={VTV^–1 =(VT₁V^–1,...,VT_nV^–1): V is invertible onxxHx}. We study the structure of the norm closure ofxxSx, both in the case when T is commutative and when it is not. We first develop a Rota-model for the Taylor spectrum and use it to study n-tuples with totally disconnected Taylor spectrum, in particular quasinilpotent ones. We then consider limits of nilpotent n-tuples, and of normal n-tuples. For noncommuting n-tuples, we present a number of surprising facts relating the closure ofxxSx(T) to the Harte spectrum of T and the lack of commutativity of T. We show that a continuous function which is constant onxxSx(T) for all T must be constant. We conclude the paper with a detailed study of closed similarity orbits.Research partially supported by grants from the National Science Foundation.     

On the similarity of dual operators

Let k be a field with an involution σ and a non-degenerate sesquilinear form, where V,W are n-dimensional k-spaces. Assume that ΛEnd(V) and Λ^*End(W) are dual operators. We show that if Λ and Λ^* are similar, then Λ^*=Λ^-1, where :V→W is Hermitian.     

On similarity of quasinilpotent operators

Bounded linear operators on separable Banach spaces algebraically similar to the classical Volterra operator V acting on C[0,1] are characterized. From this characterization it follows that V does not determine the topology of C[0,1], which answers a question raised by Armando Villena. A sufficient condition for an injective bounded linear operator on a Banach space to determine its topology is obtained. From this condition it follows, for instance, that the Volterra operator acting on the Hardy space Hp of the unit disk determines the topology of Hp for any p∈[1,∞].     

On the similarity to self-adjoint operators

An approach to the similarity problem is presented, which is based on the notion of a w-perturbation of the Volterra operator and uses the theory of Muckenhoupt weights.     

On certain locally homogeneous Clifford manifolds

Given a manifoldM, a Clifford structure of orderm onM is a family ofm anticommuting complex structures generating a subalgebra of dimension 2 ^m of End(T(M)). In this paper we investigate the existence of locally invariant Clifford structures of orderm2 on a class of locally homogeneous manifolds. We study the case of solvable extensions ofH-type groups, showing in particular that the solvable Lie groups corresponding to the symmetric spaces of negative curvature carry invariant Clifford structures of orderm2. We also show that for eachm and any finite groupF, there is a compact flat manifold with holonomy groupF and carrying a Clifford structure of orderm.Partially supported by Conicor (Argentina)Partially supported by grants from Conicet, Conicor, SECYTUNg (Argentina), and I.C.T.P. (Trieste)Partially supported by grants from Conicet, Conicor, SECYTUNC (Argentina), T.W.A.S and I.C.T.P. (Trieste)     

On non-normalizable orbits of isometric actions on Lorentz manifolds

For an isometric action of a Lie group G on a Lorentz manifold (M, g) we consider non-normalizable orbits, i.e. orbits which do not posses a G-invariant normal bundle. Orbits of this type are lightlike. It is shown, that such orbits contain lightlike homogeneous geodesics. Moreover, conditions are given, under which there exists a set of normalizable orbits having an open dense union.     

On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem

In this paper we show for most choices of masses in the coplanar n-body problem and for certain masses in the three-dimensional n-body problem that the set of initial conditions leading to complete collapse forms a smooth submanifold in phase space where the dimension depends upon properties of the limiting configuration. A similar statement holds for completely parabolic motion. By a proper scaling, these two motions become dual to each other in the sense that one forms the unstable set while the other forms the stable set of a particular set of points in the new phase space. Most of the paper is devoted to solving the Painleve-Wintner problem which asserts that these types of orbits cannot enter in an infinite spin.     

On pointwise and analytic similarity of matrices

LetA(ε) andB(ε) be complex valued matrices analytic in ε at the origin.A(ε)≈ _p B(ε) ifA(ε) is similar toB(ε) for any |ε|<r,A(ε)≈_a B(ε) ifB(ε)=T(ε)A(ε)T ⁻¹(ε) andT(ε) is analytic and |T(ε)|≠0 for |ε|<r! In this paper we find a necessary and sufficient conditions onA(ε) andB(ε) such thatA(ε)≈ _a B(ε) provided thatA(ε)≈ _p B(ε). This problem arises in study of certain ordinary differential equations singular with respect to a parameter ε in the origin and was first stated by Wasow. Sponsored by the United States Army under Contract No. DAAG29-75-C-0024     

On operators which commute with analytic Toeplitz operators modulo the finite rank operators

It is shown that an operator on the Hardy space (or ) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if . Here is a finite rank operator, and in the case , is a sum of a rational function and a bounded analytic function, and in the case , is a bounded analytic function.

     

On operators with closed analytic core

As shown by Mbekhta [9] and [10], the analytic core and the quasi-nilpotent part of an operator play a significant role in the local spectral and Fredholm theory of operators on Banach spaces. It is a basic fact that the analytic core is closed whenever 0 is an isolated point of the spectrum. In this note, we explore the extent to which the converse is true, based on the concept of support points. Our results are exemplified in the case of decomposable operators, Riesz operators, convolution operators, and semi-shifts.