$(1)$ If $T \in B(H)$ is self-adjoint and $T \neq 0$ then $T^n \neq 0$ $(a)$for $n=2,4,8,16,... (b)$ for every $n$
$(2)$ Show that any $T \in B(H)$ can be uniquely expressed as $T=T_1+iT_2$ where $T_1$ and $T_2$ are self-adjoint.
$(3)$ If $T$ is self-adjoint operator on $H$.show that $$ \|T\|= \sup \bigg\{ \frac{|\langle Tx,x\rangle|}{\|x\|^2}:x\neq 0 \bigg\}=a$$
$(4)$ Show by an example composition of two positive operator need not positive operator.
$(5)$ show that an isometric linear operator $T:H \to H$ which is not unitary maps the Hilbert space H onto a proper closed subspace of H.
I tried $(3)$ I already proved that $a \leq \|T\| $
In $(4)$ I know that the linear map represented by the matrix
$ \begin{pmatrix} a & b\\ c & d\\ \end{pmatrix}$
where $a,b,c,d \in \mathbb{R}$ $T$ is positive iff $b=c$, $a\geq 0$,$b \geq 0$ and $ad \geq b^2$ but I am fails to find such matrix.Is it possible? In $(1),(2)$ and $(5)$ I have no idea so please give me hints and some explanation for all these questions.Please help me.Thanks in Advance.