Operators with an absolute functional calculus

We study sectorial operators with a special type of functional calculus, which we term an absolute functional calculus. A typical example of such an operator is an invertible operator A (defined on a Banach space X) considered on the real interpolation space (Dom(A), X) _θ,p where 0 < θ < 1 and 1 < p < ∞. In general the absolute functional calculus can be characterized in terms of real interpolation spaces. We show that operators of this type have a strong form of the H ^∞-calculus and behave very well with respect to the joint functional calculus. We give applications of these results to recent work of Arendt, Batty and Bu on the existence of Hölder-continuous solutions for the abstract Cauchy problem.     

A solution to the problem of Lp-L^p-maximal regularity

We give a negative solution to the problem of the -maximal regularity on various classes of Banach spaces including -spaces with . Received June 11, 1999; in final form September 6, 1999 / Published online September 14, 2000     

The $H^{\infty}-$calculus and sums of closed operators

We develop a very general operator-valued functional calculus for operators with an calculus. We then apply this to the joint functional calculus of two sectorial operators when one has an calculus. Using this we prove theorem of Dore-Venni type on sums of sectorial operators and apply our results to the problem of maximal regularity. Our main assumption is the R-boundedness of certain sets of operators, and therefore methods from the geometry of Banach spaces are essential here. In the final section we exploit the special Banach space structure of spaces and spaces, to obtain some more detailed results in this setting. Received: 3 July 2000 / Revised version: 31 January 2001 / Published online: 23 July 2001     

Compact convex sets and complex convexity

We construct a quasi-Banach space which cannot be given an equivalent plurisubharmonic quasi-norm, but such that it has a quotient by a one-dimensional space which is a Banach space. We then use this example to construct a compact convex set in a quasi-Banach space which cannot be affinely embedded into the spaceL ₀ of all measurable functions.     

Uniqueness of unconditional bases in quasi-banach spaces with applications to Hardy spaces

We prove some general results on the uniqueness of unconditional bases in quasi-Banach spaces. We show in particular that certain Lorentz spaces have unique unconditional bases answering a question of Nawrocki and Ortynski. We then give applications of these results to Hardy spaces by showing the spacesH _p (T ⁿ ) are mutually non-isomorphic for differing values ofn when 0<p<1. The research of the first two authors was partially supported by NSF-grant DMS 8901636.     

Set-functions and factorization

Both authors were supported by grants from the National Science Foundation     

Subseries convergence in topological groups and vector spaces

LetG be a separable complete metric additive topological group; it is shown that if a series Σx _n is subseries convergent in any weaker Hausdorff group topology onG, then Σx _n converges inG. This result can be used to obtain various extensions of the classical Orlicz-Pettis Theorem on subseries convergence in locally convex spaces.