I was wondering about inner product spaces and adjoints. The more I think about this the more I seem to be confusing myself.
I am not sure if given T and S as operators and an inner product space if the following are true:
A) $\langle Tx,Tx\rangle = \langle x,T^*Tx\rangle$ or $\langle x,TT^*x\rangle$
B) $\langle Tx,Tx\rangle = \langle T^*Tx,x\rangle$ or $\langle TT^*x,x\rangle$
Does anything still hold if we have a $y \neq x$?
C) $\langle Tx,Ty\rangle = \langle x,T^*Ty\rangle$ or $\langle x,TT^*y\rangle$
D) $\langle Tx,Ty\rangle = \langle T^*Tx,y\rangle$ or $\langle TT^*x,y\rangle$
Does anything still hold if we have a $S \neq T$?
E) $\langle Tx,Sx\rangle = \langle x,T^*Sx\rangle$ or $\langle x,ST^*x\rangle$
F) $\langle Tx,Sx\rangle = \langle TS^*x,x\rangle$ or $\langle S^*Tx,x\rangle$
G) $\langle Tx,Sy\rangle = \langle x,T^*Sy\rangle$ or $\langle x,ST^*y\rangle$
H) $\langle Tx,Sy\rangle = \langle TS^*x,y\rangle$ or $\langle S^*Tx,y\rangle$
I am sorry if this is a basic question, I just can't seem to find it in my book.
The adjoint $A^*$ of a linear map $A:U\to V$ satisfies (is defined by) the following equation, for all $u\in U, \, v\in V$ $$\langle v, Au\rangle=\langle A^*v, u\rangle$$ Using symmetry of the inner product (and taking complex conjugates in the complex case), this also means $$\langle Au, v\rangle=\langle u, A^*v\rangle$$ Applying these properly answers each of your questions. E.g. take $A=T, \ u=x, \ v=Tx$ for A), B)