# Adjoints : the mysterious linear operator!

The definition of Adjoints says "Let $T$be a linear operator on an inner product space $V$. Then we say that $T$ has an adjoint on $V$ if there exists a linear operator $T^*$ on $V$ such that $\langle Ta|b\rangle=\langle a,|T^* b\rangle$ for all $a,b$ in $V$.

I believe every linear map can be visualized geometrically [at least for low dimensional cases]. According to Halmos "The concept of duality was defined in terms of very special linear transformations, namely linear functionals; and it has a lot to do with more general linear transformation"

suppose $A$ is an arbitrary linear transformation on a vector space $V$ and $u$ is an arbitrary element of the dual space $V'$. Now the $u(Ax)$ depends on $x$ alone [$u,A$ regarded temporarily fixed]. In other words, a function $v$ on $V$ is defined by $v(x)=u(Ax)$ then $v$ is a linear functional, an element of $V'$. we can think this as - The new element $v$ can be viewed as the result of operating on the old element $u$ by a transformation $A'$, so that $v=A'u$. The Transformation $A'$ is called the adjoint of $A$; it sends vectors in $V'$ to vectors in $V'$.

My questions are-

$1$. Is there any geometrical way of thinking about Adjoints?

$2$. what does Hoffman-Kunze mean here- "The mapping $T\rightarrow T^*$ is a conjugate-linear anti-isomorphism of period 2"

• – Rahul Apr 2 '18 at 8:01
• Anti-homomorphism means $(AB)^* = B^* A^*$, and period 2 means that $(A^*)^* = A$. – Nick Apr 7 '18 at 2:12