Multiplication operator and trace class

Question

Suppose we work in $H=l^2(\Bbb{N})$ and suppose the multiplication operator $T_f$ such that $T_f\psi=f\psi$ and $f:\Bbb{N}\rightarrow \Bbb{C}$. We denote by $B_1(H)$ the trace class of operators.

Question: I want to find a sufficient and necessary condition for $f$, such that $T_f\in B_1(H)$.

Can someone help me with this question? (This a question from a exam for Introduction to Functional analysis.)

Answer 1

The adjoint is given by $T_{\bar f}$ hence $(T_f^*T_f)^{1/2}\psi=|f|\psi$. By definition, $T_f$ is trace-class if and only if the sequence $\{f(n)\}$ is summable, i.e. in $\ell^1$.

Multiplication operators are interesting in the following sense. With the notations of the OP,

• $T_f$ is bounded if and only if the sequence $f$ is bounded;
• $T_f$ is compact if and only if $f(n)\to 0$;
• $T_f$ is trace-class if and only if $f\in\ell^1$;
• $T_f$ is Hilbert-Schmidt if and only if $f\in\ell^2$.