Ebook: Norm Ideals of Completely Continuous Operators

Contents:

Preliminaries and notation. . .
1. The class of operators of the form 1: j Aj Cfj @ 1pj 7
1. The opera tor 1: j Aj Cfj @ 'lfJj ' . . . . . 7
2. The spectrum of the operator 1: j Aj Cfj @ (fj 10
3. Completely continuous operators . . . . . 12
4. The spectral representation of a completely continuous Hermitean
operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5. Some equalities and inequalities for the proper values of completely
continuous Hermitean operators. . . . . . . . . . 21
6. Some ideals of operators . . . . . . . . . . . . . . . . . . . . 23
7. The ideals of completely continuous operators ...... . 25
8. Some uniformly closed left ideals of completely continuous operators 27
II. The Schmidt-class . . . . . . . . . . . . . . . . . . . 29
1. (a c) as a Hilbert space of completely continuous operators 29
2. The Schmidt-class of operators on L2. 34
III. The trace-class . . . . . . . . . . . . 36
1. (T c) as a Banach space of completely continuous operators 36
2. A maximum problem in ('r c) . . . . . . . . . . . . . 43
IV. The successive conjugate spaces of the space operators . . . . . . . . . . . .. .... 45
1. A characterization of 2. <£ is not a conjugate space . . .. ..... 48
3. Some linear functionals on the space of all operators 50
V. Norm ideals. . . . . . . . .
1. Crossnorms and norm ideals.
2. A maximum problem for
n
3. Symmetric gauge functions on
n
4. The unitarily invariant crossnorms on 9n .
5. The symmetric gauge functions on
and the unitarily invariant cross-
norms on 9 ...... . . . . . . . . . . . . .
6. A special class of symmetric gauge functions . . . . .
7 . Norm ideals and th e minimal nonn ideals . . . . . .
8. The uniqueness of norm for the minimal norm ideals. .
9. An open problem
Bibliography ......