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

ISBN-10: 1483229742

ISBN-13: 9781483229744

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Of degenerate operators. N o w we prove that the space of nuclear operators is the completion of the space of degenerate operators in the trace norm. F o r this it is suflScient to show that the space of nuclear operators is complete relative to the trace norm. First we shall prove the following theorem. T h e o r e m 10. L e t A^^ ··· be a sequence of nuclear operators such that the set {|| A^^ | | i } of trace norms is bounded. 4 Operators of Hilbert-Schmidt T y p e 51 Proof. F r o m the boundedness of the set {|| A^^ Wy} follows the existence of a number Μ such that X\{AnUh)\^M k=l for all orthonormal systems { / ^ and { Ä J in and H2 and every operator Αγ^.

In fact, for any vector fe Hi one has the inequality {BfJ) = {A*Af,f)^{Af,Af)^0. Consequently, as was stated above, the operator Β has the form Be^ = λ^€^, where ^i, eg, ... is an orthonormal basis in i/^, > 0, and lim^_,oo Κ = 0. We now introduce a new operator Τ = B^ defined by Τβγ^ = VX^e^. Obviously, = B. Moreover, it is clear that Τ is completely continuous and positive definite. Let us compare || Af\\ and || Γ / | | . We have II AfW^ = {Af, Af) = {A^Afyf) = {TJJ). But Τ is positive definite and consequently self-adjoint.

2 . 4 . The Trace N o r m In this section it will be proved that the nuclear operators form a linear space and that this space is the completion of the space of degenerate operators relative to a certain norm, called the trace norm. First we prove the following theorem. T h e o r e m 9. ^4 is a nuclear operator m a p p i n g the Hilbert space Hi into the Hilbert space H^y then SUpX|(^/n,^n)l=XA„ n=-l (14) n=l where the are the eigenvalues of the positive-definite operator Τ appearing in the decomposition A = UT, and the s u p r e m u m is taken over all orthonormal systems of vectors { / ^ } and {^^} in the spaces Hi and H^.

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

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