Adjoint Operator

**everk** · May 24th 2011, 08:13 AM

Hi!!!
I don't know how to solve this.
Let $V$ a inner product space, $T: V \rightarrow V$ a linear aplication and $T^*$ its adjoint.
Prove that $ker(TT^*+T^*T) = ker(T)\cap ker(T^*)$
Obviously there is an easy inclusion
Thanks!!
Everk

**girdav** · May 24th 2011, 08:32 AM

Let $x\in V$ such that $T^*Tx+TT^*x =0$ . We have $0\leq \langle Tx,Tx\rangle =\langle T^*Tx,x\rangle = -\langle TT^*x,x\rangle =-\langle T^*x,T^*x\rangle\leq 0$ .

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