# Is there a such thing as an operator of operators in mathematics?

Thus far I have seen operators of numbers and operators that perform on functions like Laplace, Fourier and Z-Transforms but is there an operator in existence that performs on other operators?

Like a trasform of a transform so to speak?

• Sounds like you're looking for the automorphism group of an operator group. – Alexander Gruber Apr 25 '13 at 12:54
• How is an operator different from a function? – rschwieb Apr 25 '13 at 13:15
• I know what you mean @rschwieb and the answer is an operator is a function just as a transform is a function of functions. Perhaps I mean a transform of transforms. – Nirma Apr 25 '13 at 13:21
• @AlexanderGruber I think that is what Im looking for. – Nirma Apr 25 '13 at 13:22
• @Nirma I think you are under the impression that there is a careful naming scheme for using words like "transform" and "operator." There isn't, really. They are all just functions, and functions can have any sets for their domain and codomains. The only reason we bring in special words like "transform" "homomorphism" and "operator" is partially so that we're not using just "function" all the time and partially disciplinary custom. – rschwieb Apr 25 '13 at 13:41

## 1 Answer

Look at some of the hits from a google search for "exponentiation of operators". Among other things, you'll see that exponentials of various differentiation operators, e.g. $e^{\frac{d}{dx}}$, arise in quantum mechanics. More general than exponentiation of operators are operators of operators defined by various power series expansions of an operator.

Also, searches for operator algebras, and especially for maps on operator algebras, will bring up things you're looking for. Below are a couple of examples I found without looking very deep into these searches:

Robert T. Moore, Exponentiation of operator Lie algebras on Banach spaces, Bulletin of the American Mathematical Society 71 #6 (November 1965), 903-908.

Jinchuan Hou, Additive maps on standard operator algebras preserving invertibilities or zero divisors, preprint, not dated, 16 pages.