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A Problem in Linear Matrix Approximation
Author(s) -
Berens H.,
Finzel M.
Publication year - 1995
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19951750104
Subject(s) - mathematics , hilbert space , linear subspace , subspace topology , commutator , norm (philosophy) , matrix norm , low rank approximation , pure mathematics , combinatorics , operator (biology) , mathematical analysis , discrete mathematics , algebra over a field , eigenvalues and eigenvectors , biochemistry , lie conformal algebra , physics , chemistry , repressor , quantum mechanics , tensor (intrinsic definition) , political science , transcription factor , law , gene
Let A be a normal operator in ℬ(H), H a complex Hilbert space, and let ℛ A = ≷ { AX ‐ XA:X ∈ ℬ ( H )} be the commutator subspace of ℬ( H ) associated with A . If B in ℬ( H ) commutes with A , then B is orthogonal to ℛ A with respect to the spectral norm; i.e., the null operator is an element of best approximation of B in ℛ A . This was proved by J. Anderson in 1973 and extended by P. J. Maher with respect to the Schatten p ‐norm recently. We take a look at their result from a more approximation theoretical point of view in the finite dimensional setting; in particular, we characterize all elements of best approximation of B in R A and prove that the metric projection of H onto ℛ A is continuous.
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