Unusual traces on operator ideals. (English) Zbl 0653.47025
Let H be a separable Hilbert space and B(H) (resp. \(B_ 0(H))\) the space of all bounded linear operators (resp. compact operators) defined on H. Given an ideal J in \(B_ 0(H)\) a trace on J is a linear functional \(\tau\) on J so that
(1) \(\tau (P)=1\) if P is a rank one projection,
(2) \(\tau (XY)=\tau (YX)\) if \(X\in J\) and \(Y\in B(H).\)
We say that \(\tau\) is separately continuous if
(3) For every \(X\in J\), the linear functional \(Y\to \tau (XY)\) is bounded on B(H).
The functional \(\tau\) coincides with the usual trace on finite rank operators. A positive trace class operator T is said to be uniquely traceable if the only separately continuous trace defined on the ideal generated by T is the usual trace.
An ideal J is said to be quasi-normed if there is a quasi-norm \(| \cdot |\) on J with respect to which J is complete and satisfies
(a) \(| X| \geq \alpha \| X\|\), \(X\in J\) for some \(\alpha >0\), and
(b) \(| YXZ| \leq \| Y\| | X| \| Z\|\), \(Y,Z\in B(H)\), \(X\in J.\)
The article under review deals with quasi-Banach oerator ideals which admit more than one continuous trace.
Ideals are constructed which support a large amount of different continuous traces. For an alternative approach to Kalton’s example and some historical background see the Appendix of A. Pietsch [Eigenvalues and s-numbers (1987; Zbl 0615.47019)].
(1) \(\tau (P)=1\) if P is a rank one projection,
(2) \(\tau (XY)=\tau (YX)\) if \(X\in J\) and \(Y\in B(H).\)
We say that \(\tau\) is separately continuous if
(3) For every \(X\in J\), the linear functional \(Y\to \tau (XY)\) is bounded on B(H).
The functional \(\tau\) coincides with the usual trace on finite rank operators. A positive trace class operator T is said to be uniquely traceable if the only separately continuous trace defined on the ideal generated by T is the usual trace.
An ideal J is said to be quasi-normed if there is a quasi-norm \(| \cdot |\) on J with respect to which J is complete and satisfies
(a) \(| X| \geq \alpha \| X\|\), \(X\in J\) for some \(\alpha >0\), and
(b) \(| YXZ| \leq \| Y\| | X| \| Z\|\), \(Y,Z\in B(H)\), \(X\in J.\)
The article under review deals with quasi-Banach oerator ideals which admit more than one continuous trace.
Ideals are constructed which support a large amount of different continuous traces. For an alternative approach to Kalton’s example and some historical background see the Appendix of A. Pietsch [Eigenvalues and s-numbers (1987; Zbl 0615.47019)].
Reviewer: K.Seddighi
MSC:
| 47L30 | Abstract operator algebras on Hilbert spaces |
| 47B10 | Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.) |
Keywords:
trace on finite rank operators; positive trace class operator; quasi- Banach oerator ideals which admit more than one continuous traceCitations:
Zbl 0615.47019References:
| [1] | Operator Ideals, North-Holland, Amsterdam 1979. |
| [2] | Pietsch, Math. Nachr. 100 pp 61– (1981) |
| [3] | Eigenvalues and s-numbers, Mathematik und ihre Anwendungen in Physik udn Technik, Band 43, Leipzig 1987 |
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