# Adjoint of canonical expansion of compact operator

Lets say I have given a rank-$n$ operator $A = \sum^n_{k=1} \lambda_k \langle u_k, \cdot \rangle v_k$. Then it is straightforward to compute its adjoint as $A^\ast = \sum^n_{k=1} \lambda_k \langle v_k, \cdot \rangle u_k$.

Now, if $A$ is a compact operator on a Hilbert space, then $A$ has a canonical representation of the form $A = \sum_k \lambda_k \langle u_k, \cdot \rangle v_k$, where $\{u_k\}$ and $\{v_k\}$ are orthonormal systems and the $\lambda_k$ are the singular values (i.e. the eigenvalues of $A^\ast A$).

Just as in the finite rank case, I could formally deduce from this canonical form for $A$ a canonical form for $A^\ast$: $$A^\ast = \sum_k \lambda_k \langle v_k, \cdot \rangle u_k$$ Is there anything i have to regard more carefully because i am handling an infinite expansion?

No. Since $\lambda_k\to 0$ you have a norm-convergent expansion. The $*$-operation is norm-continuous.