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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Applying L e m m a 4 to the operators B'^A and B, we find that '^{B-^ABe„,e„) = f^{Ae„,e„). 44 T H E KERNEL THEOREM Ch. I T h u s we have shown that 00 00 n=l Because the right side of the equation does not depend upon the choice of basis { / ^ } , the value of Σ ^ ^ Χ ( 4 / n » ^ n ) ^^^s not depend upon the choice of unconditional basis { / ^ } , which proves the theorem. Somewhat more careful considerations show that the following theorem, proven by V. B. Lidskii, is valid. T h e o r e m 6 {On the Trace).

In the space H^, Proof. L e t / i , / 2 , ... be an orthonormal basis in the space and suppose that the series Σ ^ ^ χ || 4/n II converges. T h e n the series Σ η = ι Ι Ι 4/n IP also converges. Therefore ^ is a H i l b e r t - S c h m i d t operator and a fortiori complete continuous. Consequently A can be represented in the form A = UT. Since ||/^ || = 1 and the operator U is isometric on the range of Γ , we have σ/η,/η) < II ^/n II = 11 Ί / Γ / J I = M / J | . T h i s inequality shows, in view of the convergence of Σ * « ι || 4/n \l ^bat the series Σ * = ι {Tf^^^f^) converges.

Io Proof. L e t Γ be a positive-definite nuclear operator. We introduce the operator T'^ setting T^e^ = λ*^^, where (^^} is an orthonormal * I f II Omn II i s t h e m a t r i x c o r r e s p o n d i n g t o t h e o p e r a t o r A w i t h r e s p e c t t o t h e b a s i s { / „ } , then {Af„,f„) = Σ^^^α„„ a n d i s t h u s t h e t r a c e o f t h e m a t r i x || Omn \l T h e a n a l o g o u s a s s e r t i o n , w i t h o u t t h e a s s u m p t i o n o f t h e c o m p l e t e c o n t i n u i t y o f T, is p r o v e d b e l o w i n T h e o r e m 7.

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