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On the length of faithful nuclear representations of finite rank operators. (English) Zbl 0654.47007

If T is a finite rank operator between Banach spaces, a nuclear representation for T is a series representation of T as an absolutely convergent sum of rank one operators, with the nuclear norm of T being the infimum over all such representations of the sum of the norms of the rank one operators. The length of a representation is the number of terms in the sum, and the representation of T is called faithful if the sum of the norms of the terms in the sum equals the nuclear norm of T.
This paper is a study of the length of faithful nuclear representations of operators between finite dimensional spaces. For example, it is known that if E and F are real spaces of dimension m and n, resp., then any T:F\(\to E\) has a faithful representation of length \(N=mn\). It is shown here that this result is exact in the sense that if \(k\leq N\) there are spaces E and F and an operator T:F\(\to E\) for which the minimal length of a faithful nuclear representation is k. Again, it is shown that if F is of dimension n with the property that whenever dim E\(=n\) then every T:F\(\to E\) has a faithful representation of length n, then F is isometric to \(\ell_ n^{\infty}\). Other results of a similar nature are also given.
Reviewer: J.R.Holub

MSC:

47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
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References:

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