Back in the day you used to have to connect to an operator to call anyone. Instead of dialing the number yourself (or just pressing the person's name on your iPhone), you called the operator, who then connected you. If you ever watch a sitcom from the '50s or '60s, you can see this in action.
When Adam tries to call Amy Hertz, he connects through an operator. What's strange is that it seems to be the same operator every time, even though he's calling from different locations. It's a man's voice and Adam notes that it's "startling to hear a man's voice on the line" (9.9) (because most telephone operators were women).
What's the point of having the operator be male? Maybe we're supposed to sense that something is up. Or maybe the operator is actually Luke, the switchboard operator at the hospital. This is one of those times where we're not sure what's real and what's imagined. It's possible that Adam is actually making phone calls from the hospital, in which case the operator might be Luke (or at least it makes sense that it's the same person every time). Or Adam might be imagining the whole thing. Either way, upon reflection the reader can definitely tell that something strange is going on.