Linearity of A Hermitian Operator

[buraqenigma]

Can anybody give me a hint about how can i show that if an operator is linear then it's hermitian conjugate is linear. Thanks for your help from now.

 

[CompuChip]

I don't know the context, but if you have an inner product perhaps you can try to show that
$$\langle \psi | T^* (\alpha \phi + \beta \psi) \rangle = \langle \psi | \alpha T^* \phi + \beta T^* \psi \rangle$$
for any $\psi, \phi, \chi \in \operatorname{domain} T$, which would prove the linearity of $T^*$?

 

[buraqenigma]

i want to say that how can i show if $$A$$ is linear , $$A^\dagger$$ is linear.i don't know where can i start.please help.

 

[dextercioby]

2 questions for the OP:
* How do you define the domain of definition of the adjoint of a (possibly unbounded) densly defined linear operator in a Hilbert space ?
* What is the definition of linearity for an unbounded operator in a Hilbert space ?

I asked these 2 qtns because we want to the give the proof in the most general case, namely when the operator is unbounded but linear and densly defined.

And btw, this is a purely mathematical problem, it has nothing to do with quantum mechanics.

 

[buraqenigma]

Explaining

if A is lineer operator

A[af(x)+bg(x)]=aAf(x) + bAg(x)

x is the parameter of functions in Hilbert space

 

[dextercioby]

Who's "x" ?

 

[CompuChip]

And how is $A^\dagger$ defined?