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Compressions of linearly independent selfadjoint operators
Rodman, Leiba Spitkovsky, Ilya M.
Rodman, Leiba
Spitkovsky, Ilya M.
Abstract
The following question is considered: What is the smallest number gamma(k) with the property that for every family {X-1,..., X-k} of k selfadjoint and linearly independent operators on a real or complex Hilbert space H there exists a subspace H-0 subset of H of dimension gamma(k) such that the compressions of X-1,..., X-k to H-0 are still linearly independent? Upper and lower bounds for gamma(k) are established for any k, and the exact value is found for k = 2, 3. It is also shown that the set of all gamma(k)-dimensional subspaces H-0 with the desired property is open and dense in the respective Grassmannian. The k = 3 case is used to prove that the ratio numerical range W(A/B) of a pair of operators on a Hilbert space either has a non-empty interior, or lies in a line or a circle. (C) 2011 Elsevier Inc. All rights reserved.
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2012-01-01
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Mathematics
DOI
10.1016/j.laa.2011.10.038