A Problem in Linear Matrix Approximation

✍ Scribed by H. Berens; M. Finzel

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 560 KB
• Volume: 175
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19951750104

✦ Synopsis

Abstract

Let A be a normal operator in ℬ︁(H), H a complex Hilbert space, and let ℛ ~A~ = ≷ {AX ‐ XA:X ∈ ℬ︁(H)} be the commutator subspace of ℬ︁(H) associated with A. If B in ℬ︁(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.