Kernel of the adjoint of a linear operator

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I think I found out how to do this. Are my steps right?
Let x be an element in the kernel of T. Since TT*x = T*Tx = T*0=0, (TT*x,x) = 0 where (,) indicates the inner product.
Then, (TT*x,x) = (T*x,T*x) = ||T*x||^2 = 0 so T*x = 0 and x is in the kernel of T*. The other way can be done similarly.

BTW, what does "x*" indicate in x*(T*T)x? I don't understand...

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