Two modest remarks. Firstly one would need to know on which spaces (presumably suitable $L^2$-spaces) your spaces are acting on and what regularity or growth properties you are demanding from the kernels, in order to attack the problem rigorously. Secondly, operators with triangular kernels tend to be quasi-nilpotent and so anintegral operator which cannot be expressed as product of two such operators would be counter-example.

@AlexanderShamov. no, it's not.

unbounded normal operators. Where did Fourier series and spherical functions come from if not from here and where would mathematics be without them? The potential usefulness of a parametrisation as in the OP in the presence of central symmetry seems pretty obvious to me and, of course, it is the exploitation of this and more sophisticated symmetries which lie at the centre of the above and many more key mathematical theories and concepts.

With respect, I have to disagree strongly with the first comment, especially the final sentence thereof. Surely one of the BIG themes of mathematics is the use of special frames for specific problems---think canonical forms. Examples: the Jordan canonical form, the spectral theorem for normal matrices, bounded and

Since you mention spirals, a really remarkable family of these are the so-called McLaurin spirals which were discovered by this Scottish mathematcian in the 18 th century. They are those with $f$ of the form $(\cos(n \theta))^{\frac 1n}$ and have a plethora of remarkable properties, e.g., they are all orbits for power laws. The logarithmic spiral is a (degenerate) special case. They are also catenaries for such laws.

I don't know of one but there is a standard way to go from one to higher dimensions. Thus in going up to dimension 2 one fixes the second variable and uses the given conditions on the proposed set of uniqueness to deduce that the corresponding function of one variable is identically zero. One then iterates.

There are whole books on this. I would suggest "Entire Functions" by Ralph Boas for starters.

A set of uniqueness for a family of (say holomorphic) functions is a subset of their domain which is such that if a function from this family vanishes on this set, then it is identically zero.