Applications of Harmonic Analysis by I. M. Gel'fand and N. Ya. Vilenkin (Auth.)

By I. M. Gel'fand and N. Ya. Vilenkin (Auth.)

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We note that from this inequality there follows the useful relation \\AB\\^^\\A\\\\B\\2. (3) T h e product BA, where ^ is a continuous linear operator and Β is a Hilbert-Schmidt operator, is also a H i l b e r t - S c h m i d t operator. In fact, BA = B u t by property (1), B * is a H i l b e r t Schmidt operator, and by property (2) is of H i l b e r t - S c h m i d t type. A second application of property (1) shows that Β A is a H i l b e r t Schmidt operator. 3. Nuclear Operators An even more restrictive requirement than that the operator A be of Hilbert-Schmidt type is that it be a nuclear operator.

29) ^•=1 F r o m inequalities (26) and (29) it follows that IN Mill = inf2)iimi, and since || P,. || = . | | / , | | | | ^ , . | | , that M IMIIi = i n f g i l M I I I ^ . I I , fc=l where the infimum is taken over all decompositions Af = Σ ^ ΐ ι {ffk)gk of A into a sum of operators of rank 1. T h e infimum is achieved by any decomposition in which the f, are eigenvectors (with nonzero eigen­ values) of the operator Τ appearing in the decomposition A = UT, and gj, = Ufj,. ll, where the infimum is taken over all decompositions of the operator A into a sum of operators of rank 1.

We should remark, however, that we do not consider the concept of nuclearity in Banach spaces to be adequately justified, because such operators lack certain very important properties possessed by nuclear operators in Hilbert spaces. For example, the trace theorem (cf. T h e o r e m 6 ) is not valid for nuclear operators in Banach spaces. =l Λ·=1 does not hold. We remark that if an operator Ay mapping a Banach space E^ into a Banach space has the form Af=X{F,yf)g,y (30) where F^. e E[y gj. e E^, and the series X A IIF.

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