help me please about adjoint of operators in L1

Question

A : L₁→L₁

1) A x=( x₁, x₂,.....xn , 0,0,....)

2) A x= (λ₁ x₁ ,λ₂ x₂,.....)        |λ n|≤1 and λ n ∈ R


I need to find adjoint of operators A in given space.

But I am confusing.I know that = What is the y? I can not go any result. I am waiting help. T*y =?

Answer 1

Presumably this is $A:L^2 \to L^2$. Note that the adjoint of $A$, $A^*$, is the operator such that $\langle A x , y \rangle = \langle x, A^* y \rangle$. Write out $\langle A x , y \rangle$, and shuffle the terms out to get a linear operator applied to $y$, and this gives you $A^*$.

Answer 2

First, you should have $A:L^2\to L^2$.

1)For $x=(x_1,x_2,\ldots),y=(y_1,y_2,\ldots)\in L^2$, we have $$\langle Ax,y\rangle=\sum_{i=1}^nx_iy_i=\langle x,Ay\rangle.$$

So, $A^*=A$.

2)You can check analogously that $A^*=A$.