Small transitive families of subspaces
W. E. Longstaff (University of Western Australia)
ABSTRACT. A family of subspaces of a complex separable Hilbert space is called transitive if every
bounded operator which leaves each of its members invariant is scalar. In this talk we will survey some results
concerning transitive families of small cardinality. For example, in finite dimensions (greater than 2) the minimum cardinality of a transitive
family is 4. In infinite dimensions
- a transitive pair of linear manifolds exists,
- there is a transitive family of norm-closed subspaces with 4 elements,
- a transitive family of dense opeartor ranges with 5 elements can be found.
Also, in infinite dimensions (>1) three nest algebras (corresponding to maximal nests) can intersect in the scalar
operators, but two cannot. In infinite dimensions, it is not known if this is the case for maximal nests of
type omega+1, although four such nest algebras can intersect in the scalar operators.