Semi-groups and collectively compact sets of linear operators

Compact Sets and Compact Operators

Proof. Properties 2 and 3 are left to the reader. For property 1, assume that S is an unbounded compact set. Since S is unbounded, we may select a sequence {vn}n=1 such that ‖vn‖ → 0 as n→∞. Since S is compact, this sequence will have a convergent subsequence, say {vk}k=1, which will still be unbounded. This sequence is Cauchy, so there is a positive integer K for which ‖v`− vm‖ ≤ 1/2 for all `...

Semi-groups of Operators

G-linear sets and torsion points in definably compact groups

Let G be a definably compact group in an o-minimal expansion of a real closed field. We prove that if dim(G \ X) < dimG for some definable X ⊆ G then X contains a torsion point of G. Along the way we develop a general theory for the so-called G-linear sets, and investigate definable sets which contain abstract subgroups of G.

On Fixed Points for Semi-groups of Linear Operators

That such theorems, when applied to the space X of bounded functions on a set S, have measure-theoretic significance was noted in [3]. Let 77 be a semi-group of 1-1 transformations of S into S. To each a EH we correspond the transformation T, of X into itself where T„(f) is the function f(a(s)), sES,fEX. These transformations form a semi-group G each element of which has norm one. Furthermore G...

Weakly Compact Groups of Operators

It is shown that the weakly closed algebra generated by a weakly compact group of operators on a Banach space is reflexive and equals its second commutant. Also, an example is given to show that the generator of a monothetic weakly compact group of operators need not have a logarithm in the algebra of all bounded linear operators on the underlying space. Let X be a complex Banach space, B(X) th...