Ian Doust

BANACH SPACE OPERATORS WITH A BOUNDED H＾∞ FUNCTIONAL CALCULUS

In this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when f is holomorphic on a larger sector. We also examine how certain properties of this functional calculus, such as the existence of a bounded H ∈ functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive L p space, then T has a bounded H ∈ functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.

Enhanced negative type for finite metric trees

Abstract A finite metric tree is a finite connected graph that has no cycles, endowed with an edge weighted path metric. Finite metric trees are known to have strict 1-negative type. In this paper we introduce a new family of inequalities (1) that encode the best possible quantification of the strictness of the non-trivial 1-negative type inequalities for finite metric trees. These inequalities are sufficiently strong to imply that any given finite metric tree ( T , d ) must have strict p-negative type for all p in an open interval ( 1 − ζ , 1 + ζ ) , where ζ > 0 may be chosen so as to depend only upon the unordered distribution of edge weights that determine the path metric d on T. In particular, if the edges of the tree are not weighted, then it follows that ζ depends only upon the number of vertices in the tree. We also give an example of an infinite metric tree that has strict 1-negative type but does not have p-negative type for any p > 1 . This shows that the maximal p-negative type of a metric space can be strict.

Functional Calculus, Integral Representations, and Banach Space Geometry

Abstract On suitable Banach spaces, the existence of a certain functional calculus for an operator is equivalent to that operator having a certain integral repreaentation with respect to some family of projections. In this paper we consider the problem of identifying the precise classes of spaces on which these equivalences hold, for several of the most important functional calculi.

A COMPARISON OF ALGEBRAS OF FUNCTIONS OF BOUNDED VARIATION

Motivated by problems in the spectral theory of linear operators, we previously introduced a new concept of variation for functions defined on a non-empty compact subset of the plane. In this paper, we examine the algebras of functions of bounded variation one obtains from these new definitions for the case where the underlying compact set is either a rectangle or the unit circle, and compare these algebras with those previously used by Berkson and Gillespie in their theories of AC-operators and trigonometrically well-bounded operators.

The spectral theorem for well-bounded operators

Well-bounded operators are those which possess a bounded functional calculus for the absolutely continuous functions on some compact interval. Depending on the weak compactness of this functional calculus, one obtains one of two types of spectral theorem for these operators. A method is given which enables one to obtain both spectral theorems by simply changing the topology used. Even for the case of well-bounded operators of type (B), the proof given is more elementary than that previously in the literature.

Compact well-bounded operators

Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.

Well-bounded operators on nonreflexive Banach spaces

Every well-bounded operator on a reflexive Banach space is of type (B), and hence has a nice integral representation with respect to a spectral family of projections. A longstanding open question in the theory of well-bounded operators is whether there are any nonreflexive Banach spaces with this property. In this paper we extend the known results to show that on a very large class of nonreflexive spaces, one can always find a well-bounded operator which is not of type (B). We also prove that on any Banach space, compact well-bounded operators have a simple representation as a combination of disjoint projections.

Compact AC (σ) Operators

Abstract.All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) operator into real and imaginary parts.

The spectral type of sums of operators on non-hilbertian Banach lattices

It is known that on a Hilbert space the sum of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. It had been conjectured that this may still hold for operators on Lp spaces for p 6= 2. We show here that this conjecture is false. 1

Extensions of an AC(σ) functional calculus

On a reflexive Banach space X, if an operator T admits a functional calculus for the absolutely continuous functions on its spectrum σ(T)⊆R, then this functional calculus can always be extended to include all the functions of bounded variation. This need no longer be true on nonreflexive spaces. In this paper, it is shown that on most classical separable nonreflexive spaces, one can construct an example where such an extension is impossible. Sufficient conditions are also given which ensure that an extension of an AC functional calculus is possible for operators acting on families of interpolation spaces such as the Lp spaces.