Factorizations of idempotent operator as products of two idempotents
Tóm tắt
For two commutative idempotents
$$\Pi _1$$
and
$$\Pi _2$$
,
$$\Pi _1+\Pi _2-\Pi _1\Pi _2$$
is clearly idempotent. By studying the characterisation of two idempotents generated from factorizations of idempotent operator, we give a necessary and sufficient condition for when there is no another idempotent operator
$$\Pi _3$$
which has the same range with
$$\Pi _1\Pi _2$$
such that
$$\Pi _1+\Pi _2-\Pi _3$$
is idempotent.
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