TY - JOUR

T1 - Small transitive families of subspaces in finite dimensions

AU - Lambrou, M.S.

AU - Longstaff, William

PY - 2002

Y1 - 2002

N2 - A family F of subspaces of a finite-dimensional Hilbert space H is transitive if every operator leaving every element of F invariant is scalar. If dim H greater than or equal to 3, the minimum cardinality of a transitive family is 4. All 4-element transitive families of subspaces of 3-dimensional space are described. For spaces of dimension greater than 3, necessary, but not sufficient, conditions satisfied by every 4-element transitive family are obtained, showing that (i) either every pair of subspaces intersects in (0) or every pair spans H (but not both), (ii) at least three of the subspaces must have the same dimension (either [dim H/2] or [dim H/2] + 1), the dimension of the remaining subspace differing from this common dimension by at most 1. (C) 2002 Elsevier.Science Inc. All rights reserved.

AB - A family F of subspaces of a finite-dimensional Hilbert space H is transitive if every operator leaving every element of F invariant is scalar. If dim H greater than or equal to 3, the minimum cardinality of a transitive family is 4. All 4-element transitive families of subspaces of 3-dimensional space are described. For spaces of dimension greater than 3, necessary, but not sufficient, conditions satisfied by every 4-element transitive family are obtained, showing that (i) either every pair of subspaces intersects in (0) or every pair spans H (but not both), (ii) at least three of the subspaces must have the same dimension (either [dim H/2] or [dim H/2] + 1), the dimension of the remaining subspace differing from this common dimension by at most 1. (C) 2002 Elsevier.Science Inc. All rights reserved.

U2 - 10.1016/S0024-3795(02)00380-4

DO - 10.1016/S0024-3795(02)00380-4

M3 - Article

VL - 357

SP - 229

EP - 245

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

ER -