Intersections of nest algebras in finite dimensions

Abstract

If M, N are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n( n+1)/2, where n= dim H. For any two maximal nests M, N there exists a basis { f 1, f 2,…, f n } of H and a permutation π such that M={(0)}∪{M i:1⩽i⩽n} and N={(0)}∪{N i:1⩽i⩽n}, where M i = span{ f 1, f 2,…, f i } and N i = span{ f π(1) , f π(2) ,…, f π( i) }. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π( j)= n− j+1,1⩽ j⩽ n. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras.