Norm Ideals Of Completely Continuous Operators (ergebnisse Der Mathematik Und Ihrer Grenzgebiete. 2. Folge)

Completely continuous operators on a Hilbert space or even on a Banach space have received considerable attention in the last fifty years. Their study was usually confined to special completely continuous operators or to the discovery of properties common to all of them (for instance, that every such operator admits a proper invariant subspace). On the other hand, interest in spaces of completely continuous operators is comparatively new. Some results of this type may be found implicit in the early work of E. SCHMIDT. Other results are'generally known'and cannot be found explicitly in print. One of the interesting and relatively new results states that modulo the language of BANACH (that is, up to equivalence) the space of all operators on a Hilbert space f> is the second conjugate of the space of all completely continuous operators on f>. The study of spaces of completely continuous operators on a perfectly general Banach space involves many difficulties. Some stem, for instance, from the unsolved problem whether a completely continuous operator on a perfectly general Banach space is always approximable in bound by operators of finite rank. The answer is affirmative in all the special Banach spaces considered. An affirmative answer to the above problem is the ultimate desideratum - it ~ould simplify the theory considerably. A negative answer, however, would be equally interesting (although for us not so useful), since it would settle negatively the open'basis problem'. Cover......Page 1 Title......Page 3 Preface......Page 5 Contents......Page 7 Preliminaries and notation......Page 9 1. The operator......Page 15 2. The spectrum of the operator......Page 18 3. Completely continuous operators......Page 20 4. The spectral representation of a completely continuous Hermitean operator......Page 22 5. Some equalities and inequalities for the proper values of completely continuous Hermitean operators......Page 29 6. Some ideals of operators......Page 31 7. The ideals of completely continuous operators......Page 33 8. Some uniformly closed left ideals of completely continuous operators......Page 35 1. (a c) as a Hubert space of completely continuous operators......Page 37 2. The Schmidt-class of operators on L2.......Page 42 1. as a Banach space of completely continuous operators......Page 44 2. A maximum problem in......Page 51 1. A characterization of and......Page 53 2. C is not a conjugate space......Page 56 3. Some linear functionals on the space of all operators ......Page 58 1. Crossnorms and norm ideals......Page 62 2. A maximum problem for Rn......Page 66 3. Symmetric gauge functions on ?n .......Page 69 4. The unitarily invariant crossnorms on Rn ......Page 72 5. The symmetric gauge functions on 2 and the unitarily invariant cross-norms on R......Page 75 6. A special class of symmetric gauge functions......Page 79 7. Norm ideals and the minimal norm ideals......Page 81 8. The uniqueness of norm for the minimal norm ideals ......Page 86 9. An open problem......Page 87 Bibliography......Page 88