Definition of algebra generated by an operator.

Question

Suppose T is an operator (defined on a Hilbert space, say). Does the "algebra generated by $T$" have a commonly understood meaning? I couldn't find a satisfactory answer after some internet searching. It seems to me that there are two ways one could define the algebra $A$ generated by $T$ (without including additional structure on $A$, like closure), namely $A=\{p(T): p$ is a polynomial}, or $A=\{p(T): p$ is a polynomial with $p(0)=0$}. Both definitions contain only the operators which are linear combinations of powers of $T$. In the first definition, we allow for a power of $0$, so that the identity operator belongs to the algebra. Is this the standard definition, or is it something else?

Answer 1

The algebra generated by $T$ is usually defined to be the intersection of all algebras containing $T$. If you work out this definition, you will find that the result is one of the algebras you mention, depending on whether you require algebras to have a unit.

More genreally, if $A$ is an algebra and $S \subseteq A$ is a subset, then the subalgebra generated by $S$ is the intersection of all subalgebras $B \subseteq A$ containing $S$. Again, you have to specify whether or not you require all algebras to be unital. (On this point, different authors use different conventions.)

Answer 2

One one speaks about things generated by a set (linear subspaces, subgroups, subrings, and so on) there is usually a category in the background and then the generated object is supposed to be the smallest "acceptable" thing containing the set of generators.

If the background category is that of

• rings, you would only take polynomials without constant term,

• unital rings, then take all polynomials,

• C$^*$-algebras, then take (non commutative) polynomials in $T$ and $T^*$, with or without constant term, and close it.