The following is Exericse II.8.13 from A Course in Functional Analysis by J. B. Conway.
Show that if $T\cong T\oplus T$, then $T\cong T\oplus T\oplus\cdots$.
Here $\mathscr{H}$ is a Hilbert space, $T:\mathscr{H\to H}$ is a bounded operator, and direct sums of operators such as $T\oplus T:\mathscr{H\oplus H}\to\mathscr{H\oplus H}$ is defined coordinatewise. $T\cong T\oplus T$ means there exists a unitary isomorphism $U:\mathscr{H\to H\oplus H}$ such that $UTU^{-1}=T\oplus T$.
My question:
What does the author mean by $T\oplus T\oplus\cdots$? Should it be the direct sum of a countable number of $T$'s?
From $T\cong T\oplus T$ it is easy to deduce that $T\cong \underbrace{T\oplus\cdots\oplus T}_{\textrm{$n$ copies of $T$'s}}$. However I don't see how to deal with the countable case.
I own the book.
I think that it is actually uncountable direct sum of $T's$. If you refer back to the book, you can see that they also use the notation $\oplus_{n=1}^{\infty},$ which I believe is more standard for a countable direct sum.