Definitions of adjoints (functional analysis vs category thy)

Question

If I have a linear operator $f$ on a Hilbert space, then I define the adjoint of $f$ to be $f^*$ where,

$(fx,y)=(x,f^*y)$ for all $x,y$.

I am confused because this definitions is very different to the definition of an adjoint used in category theory.

Is there some way to tie these two definitions together?

Thanks

Matthew

Answer 1

The two notions of adjoint are related only by a symbolic similarity, enhanced by the once common but now apparently rare convention of writing $(X,Y)$ for the set of morphisms from $X$ to $Y$ (the category being understood or indicated by a subscript). Then if $L$ is left-adjoint to $R$, we have $(LX,A)\cong(X,RA)$, with obvious notational similarity to the Hilbert space definition of adjoint.