I am alittle confused about the unitary operator since I have seen many definitions
- a linear operator in an inner product space is unitary if $$\Vert T(x)\Vert =\Vert x\Vert $$
- a linear transformation on an I.P.S is said to be unitary operator if
i- T is 1-1
ii- T preserves inner product
- T is linear operator if it is invertible and preserves inner product
s0 do a 1-1 transformation need be invertible and if so what is the difference between isomorphism and unitary since I know that T is isomorphism if it is invertible and preserves inner product
For finite dimensional inner product spaces, every 1-1 operator is also an isomorphism. However, unitary operators are special isomorphisms which preserve lengths and inner products. For example, the operator which scales every vector by 2 is an isomorphism, but is not unitary.
In infinite dimensional vector spaces, not every 1-1 operator is invertible. For example, in the space $\ell^2$ of square summable sequences, the right shift operator is 1-1 but not onto; every sequence in its range begins with a zero. This operator preserves inner products, but it is not unitary because it is not invertible.