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Invariant Subspaces for Pairs of Projections
Author(s) -
Allan Graham R.,
Zemánek Jaroslav
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798006103
Subject(s) - linear subspace , invariant (physics) , mathematics , subspace topology , hilbert space , pure mathematics , projection (relational algebra) , vector space , invariant subspace , dimension (graph theory) , mathematical analysis , algorithm , mathematical physics
A simple geometrical argument shows that every pair of projections on a finite‐dimensional complex vector space has a common invariant subspace of dimension 1 or 2. The idea extends to certain pairs of projections on an infinite‐dimensional Hilbert space H . In particular every projection on H has a reducing subspace, although a finite‐dimensional one need not exist. In a final section, the results are extended to the existence of hyperinvariant subspaces for pairs of projections.