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

ISBN-10: 1483229742

ISBN-13: 9781483229744

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Extra info for Applications of Harmonic Analysis

Sample text

Are any (possibly linearly dependent) vectors in H2 and /^, . . , / ^ are any vectors in then formula (23) defines a degenerate operator. Each term {ffk)gk ^be formula is an operator which m a p s the entire space Hi onto the one-dimensional s u b s p a c e of H2 generated by the vector gj^. S u c h an operator is called an operator of rank 1. T h u s , formula (23) gives a decomposition of the degenerate operator A into a s u m of operators of rank 1. L e t us introduce the operators Pj^f = ( / , fjc)gk' T h e n E q .

In conclusion we note the following properties of H i l b e r t - S c h m i d t operators, which we will make use of further on. (1) T h e adjoint ^ * of a Hilbert-Schmidt operator is an operator of the same type. In fact, if A is of Hilbert-Schmidt type, then the series Σ * = ι || Af^ \\^ converges for every orthonormal basis { / J in Hi. But from this, by L e m m a 1, follows the convergence of the series Σ ^ = ι || for any orthonormal basis {A^} in //gthis shows that ^ * is likewise of Hilbert-Schmidt type.

F r o m this it follows that the set of degenerate operators is everywhere dense in the space of Hilbert-Schmidt operators with norm || A II2. Since this space is com­ plete, it is the completion of the set of degenerate operators in the norm || A II2. Let us pause to prove yet another property of H i l b e r t - S c h m i d t operators. Namely, we prove the following assertion. T h e o r e m 3. In order that the operator A, which takes a Hilbert space Hi into a Hilbert space H^, be of H i l b e r t - S c h m i d t type, it is necessary and sufficient that it admit a representation of the form ^/ = i)A„(/,OAn.

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

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