Given an operator $A$ in hermtian space.
Prove that either $Ax=b$ for all $b$ or $A^*y=0$ has a non-zero solution.
I tried to assume that there exist such $b$ for which there is no solution. Then I tried to use a property of adjoint operator, namely $<A^*y, b> = <y, Ab>$, but I don' t what to say further.
I think I should somehow conclude that right side is non-zero and it will imply that there exist a non-zero solution for $A^*y =0$.
You don't need $A$ to be Hermitian.
If $A$ is invertible, then $Ax=b$ has solution for all $b$ (and the solution will be nonzero whenever $b\ne0$.
If $A$ is not invertible, then neither is $A^*$, and so $A^*y=0$ has a nonzero solution.