# $T^3=\frac{1}{2}(T+T^*) \rightarrow$ T is self adjoint

Let $T$ be a normal transformation on a finite-dimensional Hilbert space; that is, $TT^*=T^*T$, where $T^*$ is the adjoint of $T$.

Prove that if $T^3=\frac{1}{2}(T+T^*)$, then $T$ is self adjoint.

I have tried to do some math on $(Tv,u)$ but I was not successful in proving the following: $(Tu,v)=(u,Tv)$ which is what I need for self-adjoint transformation.

Edit: $(T^3,v)=\frac{1}{2}\left(\left(T+T^*\right),u \right)= \left( Tu,v\right)+ \left( T^*u,v\right)=\left( u,T^*v\right)+\left( u,Tv\right)=\left( u,T^3v\right)$ Therefore, $T^3$ is self adjoint. Does it mean that $T$ is self adjoint?

Thanks!

• Are you dealing with finite-dimensional spaces, or with infinite-dimensional Hilbert spaces? Commented Feb 26, 2016 at 14:55
• That $T^3$ is self-adjoint doesn't automatically imply that $T$ is self-adjoint. But here, the equation $T^3 = \frac{1}{2}\bigl(T + T^{\ast}\bigr)$ gives more information. Suppose $\lambda$ is an eigenvalue of $T$. What does the equation tell you about $\lambda$? Commented Feb 26, 2016 at 15:00
• @DanielFischer Are you implying I should go with the direction of proving all the eigenvalues are real, therefore $T$ will be adjoint?
– Alan
Commented Feb 26, 2016 at 15:02
• If $Tv = \lambda v$, then $T^{\ast} v = \overline{\lambda} v$, if $T$ is normal. Do you denote the complex conjugate with $\lambda^{\ast}$ rather than $\overline{\lambda}$? Commented Feb 26, 2016 at 15:19
• To get $\overline{\lambda}$, you can write "\overline{\lambda}". Commented Feb 26, 2016 at 15:22

It is not true that if $T$ is normal and $T^3$ is selfadjoint, then $T$ is selfadjoint. Example: $$T=\begin{bmatrix}e^{i\pi/3}&0\\0&e^{i\pi/3}\end{bmatrix}.$$ Then $T^3=I$ (or, even when $n=1$ and $T=e^{i\pi/3}$).
But here we know more. Since $T$ is normal, it is unitarily diagonalizable. So we may assume that $T$ is diagonal. We can then see that the equality $T^3=\text{Re}\,T$ will be satisfied by the eigenvalues of $T$. If $$\tag1 \lambda^3=\text{Re}\,\lambda$$ and we write $\lambda=re^{it}$, we immediately get $$\tag2 r^3e^{3it}=r\cos t.$$ There is the obvious solution $r=0$, i.e. $\lambda=0$. For $r\ne0$, the equality $(2)$ becomes $$\tag3 r^2\cos 3t=\cos t$$ and $$\tag4 r^2\sin 3t=0.$$ The equality $(4)$ gives us $t=k\pi/3$. And now we have $$\tag5 r^2=\frac{\cos k\pi/3}{\cos k\pi}=(-1)^k\cos k\pi/3.$$For $k=0,1,2,3,4,5$, the right-hand-side in $(5)$ is $$0,-\frac12,-\frac12,1,-\frac12,-\frac12.$$ As $r^2>0$, the only acceptable values of $k$ are $0$ and $3$, which give $\lambda=1$ and $\lambda=-1$ respectively.
In summary, the only possible eigenvalues of $T$ are $0,1,-1$, and thus $T$ (being normal) is selfadjoint.
$$T^3=\frac{1}{2}(T+T^*)\Rightarrow T^4=\frac{1}{2}(T^2+TT^*)=\frac{1}{2}(T^2+T^*T)\text{ implies }TT^*=TT^*.$$
Hence T is normal. Since $$T$$ is normal, there exists an orthonormal basis consists of its eigenvectors such that $$T=UDU^*$$, where $$D$$ is diagonal and $$U$$'s column vectors are $$T$$'s eigenvectors. $$T$$ is normal also implies $$U^*(T^3)U=\frac{1}{2}(U^*TU+U^*T^*U)$$ is a real diagonal matrix.
Let $$\lambda=a+bi$$ be one of the $$T$$'s eigenvalue, then $$\lambda^3=\frac{1}{2}(\lambda+\bar{\lambda})\Rightarrow a^3-3ab^2+(3a^2b-b^3)i=a.$$ Suppose $$b\neq 0,$$ then $$b^2=3a^2\neq 0\Rightarrow 1=a^2-9a^2=-8a^2\Rightarrow a=\pm\frac{1}{2\sqrt{2}}i,$$ leads to a contradiction. Hence $$b=0$$, i.e. $$\lambda\in\mathbb{R}$$, which means $$T=T^*.$$