Strictly singular non-compact operators on hereditarily indecomposable Banach spaces

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic Banach space. The construction of this operator relies on the existence of transfinite -spreading models in the dual of the space.

     

Commutative Banach algebras and decomposable operators

For a multiplication operator on a semi-simple commutative Banach algebra, it is shown that the decomposability in the sense of Foia is equivalent to weak and to super-decomposability. Moreover, it can also be characterized by a convenient continuity condition for the Gelfand transform on the spectrum of the underlying Banach algebra. This result implies various permanence properties for decomposable multiplication operators and leads also to a useful characterization of the regularity for a semi-simple commutative Banach algebra. Finally, the greatest regular closed subalgebra of a commutative Banach algebra is investigated, and some applications to decomposable convolution operators on locally compact abelian groups are given.Research partially supported by NSF Grant DMS 90-96108.     

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X). BEJ supported by MSRVP at Australian National University; GAW supported by SERC grant GR-F-74332.     

On Hereditarily Indecomposable Banach Spaces

This paper shows that every non-separable hereditarily indecomposable Banach space admits an equivalent strictly convex norm, but its bi-dual can never have such a one; consequently, every non-separable hereditarily indecomposable Banach space has no equivalent locally uniformly convex norm.     

Semisimple Banach algebras generated by compact groups of operators on a Banach space

We prove that if τ is a strongly continuous representation of a compact group G on a Banach space X, then the weakly closed Banach algebra generated by the Fourier transforms with μ∈M(G) is a semisimple Banach algebra.     

The Banach algebras generated by operators with one-point spectrum

In this article, we present some results related to the structure of Banach algebras generated by operators with one-point spectrum.     

Interpolating hereditarily indecomposable Banach spaces

The following dichotomy is proved.

Every Banach space either contains a subspace isomorphic to , or it has an infinite-dimensional closed subspace which is a quotient of a Hereditarily Indecomposable (H.I.) separable Banach space.

In the particular case of , it is shown that the space itself is a quotient of a H.I. space. The factorization of certain classes of operators, acting between Banach spaces, through H.I. spaces is also investigated. Among others it is shown that the identity operator admits a factorization through a H.I. space. The same result holds for every strictly singular operator .

Interpolation methods and the geometric concept of thin convex sets together with the techniques concerning the construction of Hereditarily Indecomposable spaces are used to obtain the above mentioned results.     

Hereditarily hypercyclic operators and mixing

We show that hereditary transitivity (respectively strongly hereditary transitivity) is equivalent to weak mixing (respectively strong mixing) in a discrete dynamical system with Polish phase space. We also study the connection between local orbit structure and hypercyclicity, and obtain a “local hypercyclicity criterion.”     

Existence of indecomposable members in indecomposable spaces of operators

In this paper we prove that if S is a set of operators acting on a separable L _p -space X, 1 ≤ p < ∞ (or, more generally, on any separable Köthe function space) such that S is indecomposable (that is, no non-trivial subspace of X of the form L _p (A), where A is measurable, is a common S-invariant subspace), then \(\overline {span} \) S admits an indecomposable operator. As applications, we obtain some new results about transitive algebas on separable Hilbert spaces, as well as an extension of the simultaneous Wielandt theorem to semigroups of operators acting on separable L _p -spaces.     

Hereditarily indecomposable subcontinua of inverse limits of Souslin arcs are metric

We prove that if Si is a Souslin arc (a Hausdorff arc that is the compactification of a Souslin line) for each i and , then every hereditarily indecomposable subcontinuum of X is metric. Since every non-degenerate hereditarily indecomposable continuum that is an inverse limit on metric arcs is a pseudo-arc, it follows that such an X would be a pseudo-arc or a point.     

Some Characterizations of Hereditarily Indecomposable Banach Spaces

In this paper we first discuss the relations between some G-M-type spaces, and the previous eight kinds of G-M-type Banach spaces are merged into four different kinds. Then we build a Generalized Operator Extension Theorem, and introduce the concept of complete minimal sequences. Some sufficient and necessary conditions under which a Banach space is a hereditarily indecomposable space are given. Finally, we give some characterizations of hereditarily indecomposable Banach Spaces.     

On indecomposable algebras of exponent 2

For any n ≥ 3 we give numerous examples of central division algebras of exponent 2 and index 2ⁿ over fields, which do not decompose into a tensor product of two nontrivial central division algebras, and which are sums of n + 1 quaternion algebras in the Brauer group of the field. Also, for any n ≥ 3 and any field k ₀ we construct an extension F/k ₀ and a multiquadratic extension L/F of degree 2ⁿ such that for any proper subextensions L ₁/F and L ₂/F

Fredholm composition operators on algebras of analytic functions on Banach spaces

We prove that Fredholm composition operators acting on the uniform algebra H^∞(BE) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball.     

Hereditarily indecomposable subcontinua of the product of two Souslin arcs are metric

We prove that if each of X and Y is a Souslin arc (a Hausdorff arc that is the compactification of a connected Souslin line), then every hereditarily indecomposable subcontinuum of X×Y is metric.     

Hereditarily indecomposable subcontinua of the inverse limit of lexicographic arcs are metric

We show that every hereditarily indecomposable subcontinuum of the inverse limit of copies of the lexicographic arc is metric. It is observed that the technique of proof generalized to the lexicographic cube or hypercubes.     

Operators on Subspaces of Hereditarily Indecomposable Banach Spaces

We show that if X is a complex hereditarily indecomposable space,then every operator from a subspace Y of X to X is of the formI + S, where I is the inclusion map and S is strictly singular.1991 Mathematics Subject Classification 46B20, 47B99.