According to the definitions for the operator $L: H \rightarrow H$ we have:
$L$ is positive operator if the inner product $\langle Lu\mid u \rangle \geq 0$ for $\forall u \in H$
$L$ is positie-definite operator if the inner product $\langle Lu\mid u \rangle \geq \gamma \langle u\mid u \rangle$ for $\forall u \in H$ and $\gamma > 0$
It is clear that if $L$ is positive-definite so it is positive and it is stated that the opposite is not true. I am wondering why opposite does not hold, i.e. why if $L$ is positive it does not mean that $L$ is positive definite? I mean you can always find a positive $\gamma$ such that $\gamma \leq \frac{\langle Lu\mid u \rangle}{\langle u\mid u \rangle}$. Can someone give me an example of this is not true?
You can't find a positive $\gamma$ such that $\gamma \leq \frac{\langle Lu\mid u \rangle}{\langle u\mid u \rangle}$ if $\langle Lu\mid u \rangle=0$. So to find a counterexample, you might look for positive operators such that $Lu$ is sometimes $0$. The simplest example is $L=0$, the operator that sends everything to $0$.