Michael Cowling

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.

Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one

The Fourier algebra A(G) of a locally compact group G is the space of matrix coefficients of the regular representation, and is the predual of the yon Neumann algebra VN(G) generated by the regular representation of G on L 2 (G). A multiplier m of A (G) is a bounded operator on A (G) given by pointwise multiplication by a function on G, also denoted m. We say m is a completely bounded multiplier ofA (G) if the transposed operator on VN(G) is completely bounded (definition below). It may be possible to find a net ofA (G)-functions, (m i : ie I) say, such that mi tends to

H-type groups and Iwasawa decompositions☆

Since their introduction by A. Kaplan [Kpl] some ten years ago, generalised Heisenberg groups, also known as groups of Heisenberg type or H-type groups, have provided a framework in which to construct interesting examples in geometry and analysis (see, for instance, [C2], [Kp2], [Kp3], [KpR], [K2], [Rl], [R2], [TL], [TV]). The Iwasawa N-groups associated to all the real rank one simple Lie groups are H-type, so one has a convenient vehicle for studying these in a unified way: many problems on these simple Lie groups can be reduced to a problem on H-type groups, via the so-called noncompact picture, and often problems on H-type groups can be solved on all the groups of the family in one fell swoop (as in, for example, [CH], [CK], [DR]). Out of this approach to studying simple Lie groups several problems arise, such as why only some H-type groups correspond to simple Lie groups of real rank one. In this paper, we discuss various features of Iwasawa N-groups which distinguish them in the class of all H-type groups. We shall show that all H-type groups which possess certain geometric properties, clearly possessed by Iwasawa N-groups, satisfy a Lie-algebraic condition (implicit in the work of B. Kostant [Kt2]) that we shall call the J’-condition. We shall also use elementary Clifford algebra to classify the

An approach to symmetric spaces of rank one via groups of Heisenberg type

We give an elementary unified approach to rank one symmetric spaces of the noncompact type, including proofs of their basic properties and of their classification, with the development of a formalism to facilitate future computations.Our approach is based on the theory of Lie groups of H-type. An algebraic condition of H-type algebras, called J2,is crucial in the description of the symmetric spaces. The classification of H-type algebras satisfying J2 leads to a very simple description of the rank one symmetric spaces of the noncompact type.We also prove Kostant’s double transitive theorem; we describe explicitly the Riemannian metric of the space and the standard decompositions of its isometry group.Examples of the use of our theory include the description of the Poisson kernel and the admissible domains for convergence of Poisson integrals to the boundary.

Bandwidth Versus Time Concentration: The Heisenberg–Pauli–Weyl Inequality

The main result is that for quite general weight functions v, w\[ \| f \|_2 \leq K\left( {\| {vf} \|_p + \| {w\hat f} \|_q } \right)\] for all tempered distributions f for which, roughly speaking, the right side makes sense, where

A spectral multiplier theorem for a sublaplacian on SU(2)

1 \leq p

Some groups whose reduced

,

C^*

q \leq \infty

-algebra is simple

, K is a constant independent of f, and

Damping oscillatory integrals

\hat f