and the fact that it is considered "black". Don't forget that it is a model, i.e. an idealized description.

"it has to contain oscillators at all possible frequencies because it is a blackbody" -> yes this is what i mean.

well I meant that there is a single cavity containing oscillators at all possible frequencies.

well, i kind of disagree: imo, the question is quite deep and important. In my understanding, the OP is an example of situation where the presice implications in physics are intimately related to the precise mathematical understanding. For example. various QFT models utilize representations defined on the whole of the hilbert space rather than a dense subspace (which is typical of elementary QM models).

However, usually in physics, one deals with subspaces of the whole hilbert space, where the power-series expressions are valid. I was wondering, what kind of problem do you have such that the power series expressions are not valid ? it would be useful to add some context on this point.

one should consider the exponential mapping in the differential geometric sense instead

the definition of the exponential of an unbounded operator cannot be given through the power series, at least not for the whole of the space.

yes you are right on that. I was misleaded by my previous comment on the OP (which was not correct). However, i still find your argument not convincing. (Meaning that equal matrix elements do not imply that the operators are identified in general). Maybe i will try to come back later on this point with some (counter)example.

mmm yes you are right here. I was confused by the other discussion and thought OP was speaking for basis vectors. However, i still feel there are missing assumptions here (at the OP).

For example: if you consider the harmonic oscillator problem, its Fock space and the usual orthonormal basis, then the property you are describing is shared by the creation-anihillation operators: $\langle n \rvert \alpha \rvert n \rangle=\langle n \rvert \alpha^\dagger \rvert n \rangle=0$ $\forall n$. But these are certainly not equal to zero!

your "theorem" needs extra assumptions. It is not valid in this general form.