Question

Consider the linear operator T : C 3 → C 3 , defined by T (z1,z2,z3) = (z1 −iz2,iz1 +2z2 +iz3,−iz2 +z3). i) Compute T ∗ and check whether T is self-adjoint. ii) Check whether T is unitary.?

Answer 1

See below.

Explanation:

Given the rules

`z_1->z_1-i z_2 = w_1`
`z_2->i z_1+2 z_2+i z_3= w_2`
`z_3->-iz_2+z_3= w_3`

or calling

`T = ((1,-i,0),(i,2,i),(0,-i,1))`

`w=T z`

we have that `bar T = T^top`

so `T` is hermitian or self-adjoint.

but `T` is not an unitary operator because

`T cdot T^"*" ne I_3`

NOTE

`bar T` represents the `T` conjugate
`T^top` represents the `T` transpose.