inner product space question

**dannyboycurtis** · December 10th 2009, 04:10 PM

I have a question.
If T is a linear operator on an inner product space, say V, and W is a T-invariant subspace of V, how could I show that $W^{\perp}$ is T*-invariant, where T* is the adjoint of T.

**tonio** · December 10th 2009, 06:48 PM

Originally Posted by dannyboycurtis

I have a question.
If T is a linear operator on an inner product space, say V, and W is a T-invariant subspace of V, how could I show that $W^{\perp}$ is T*-invariant, where T* is the adjoint of T.

For $w'\in W^\perp$ we get:

$T^{*}w'\in W^\perp\Longleftrightarrow <w,T^{*}w'>=0\,\,\,\forall w\in W\Longleftrightarrow$ $<Tw,w'>=0\,\,\,\forall w\in W$ ...but this last equality is true....

Tonio

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