Thanks for reading! You’re right that this follows from self-adjointness. By Parseval’s identity we have

.

The last equality follows from the definition of where denotes the unit ball. Now the last expression is also equal to (by essentially the same argument just given) .

]]>sup sup [S_1 f, g]

equal to

sup sup [f, S_1 g]

It looks like some kind of self-adjointness, do we know that the operator is its own adjoint?

Thanks,

Thomas