### Inner and Outer Functions

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1.2 Some Facts from Functional Analysis 29

where A1 is an operator on M and A4 is an operator on M . This matrix representation shows why the word reducing is used.

Denition . The rank of the operator A is the dimension of its range.

Finite-rank operators (i.e., those operators whose rank is a natural num- ber) share many properties with operators on nite-dimensional spaces and thus are particularly tractable. Operators whose rank is 1 are often very useful.

Notation . Given vectors f and g H, we dene the operator f g mapping H into itself by (f g)h (h, g)f .

``Note that if neither f nor g is zero, the operator f  g has rank 1 since its``

range consists of multiples of f . Clearly, f g 0 if...

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