In functional analysis one learns that self-adjoint operators are the infinite dimensional generalisation of symmetric matrices and the dual operator is the generalisation of the transposed matrix known from linear algebra.
My question is if there is a name for linear operators $T$ on infinite dimensional spaces which satisfy $$T^{-1} = T^{*} \qquad \text{and} \qquad \| T \| = 1,$$ as orthogonal matrices do.
Best Answer
Yes. They are called unitary operators and $T^{-1}=T^*$ already implies $\|T\|=1$. In fact, such a $T$ even has to be an isometry, since
$$\|Tx\|^2=(Tx,Tx)=(T^*Tx,x)=(x,x)=\|x\|^2$$
for all $x$ (just like in the finite-dimensional case).