Toeplitz Operator is compact if and only if it has finite rank

Question

A referee has pointed out to me that it is "well known that a Toeplitz operator is compact if and only if it has finite rank" and pointed me to

R. Douglas: Banach algebra techniques in the operator theory, Academic Press, New York and London, 1972.

I have casually read through the book but I could not find this result as an explicit statement.

Is there another good reference or an accessible reasoning for this fact?

Answer 1

The explicit statement is in remark 7.15, page 182 (not sure if there is a single edition).

Corollary 7.13 implies that $\|T_\varphi\|=\inf\{\|T_\varphi+K\|:\ K\ \text{ compact }\}$. So $$ \|T_\varphi\|\leq\|T_\varphi+K\| $$ for any compact $K$. If $T_\varphi$ is compact, you can take $K=-T_\varphi$ to obtain $T_\varphi=0$.

In other words, the only compact Toeplitz operator is the zero operator.