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Fugled-Putnam’s Theorem

We shall consider matrix cases.

Let

, , valued

, w complex

Theorem

If , then

the condition C implies .,

Here the condition C is that the set of eigenvalues of is contained by .

Proof)

Now if , the we must have

two equations:

(1)

and

, (2)

which can be arranged as

(1’)

(2’)

We should note here that

Lemma We have or equivalently if and only if

.

Hence what we must show is the condition C is equivalent to that

either or

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