I'm a little confused about the definitions of positivity. I'm following Linear Algebra by Hoffmann and Kunze, and a positive operator is defined to be any operator such that $T$ is self adjoint and $<Tv,v> $ is positive for a non-zero vector $v$. The positive matrix on the other hand is a Hermitian matrix which has the property $X^{*}AX>0$ for any complex $nx1$ matrices.
Can we say any operator is positive if and only if its matrix wrt an orthonormal basis is positive?
Similarly we have defined a positive sesqui linear form $f$ which has property $f(v,v)>0$ for any non- zero $v$, and proved $f$ is positive if and only if its matrix is positive, but we also have showed that $f$ is one to one correspondence with the set of operators on $V$, so again can we link the positivity on these two things?
Yes, you can link the positivity of operator and sesqui linear form in the way that w.r.t. orthonormal basis matrix of sesqui linear form with respect to that basis is same as matrix of the operator $T$ which determines that form. If $T$ is positive then, form will be positive and correspondingly, matrices are positive.