What is an operator-valued distribution?

Question

I am a mathematician who is trying to understand Wightman axioms. I do not understand what an operator-valued distribution is, because in this context people say that operators can be unbounded, and as far as I know, unbounded operators do not have any structure on them. Indeed, if we were only dealing with bounded operators, I would guess operator-valued distributions would just be the tensor product (of vector spaces) of tempered distributions and bounded linear operators, but since this is not the case I am not really sure of what people mean.

Edits based on the received answers and comments: I do not believe the answer "it is just a set-theoretic function between the Schwartz space and unbounded operators" is satisfying because it is missing the point that it should behave in some sense like a distribution (otherwise it would not be called distribution I guess).

The reason why I am asking this is that I need to understand the maths behind it, not its meaning so I am looking for a mathematical answer from the mathematical physics community (in particular every word should be well-defined, which rules out words like abstract object or transforms, unless these words are in turn given a definition).

Answer 1

Ok, I found the answer in Folland's book. An operator-valued distribution is a function $\Phi$ from the Schwartz space to operators on $\mathcal{H}$ that has the following property: for every $\eta, \xi \in \mathcal{H}$, $\langle \eta \mid \Phi \mid \xi \rangle$ is a tempered distribution.

Just a comment for who comes next: this property is a very strong request and mathematically is the whole reason why it makes sense to call it a distribution. Without this property, things get pretty wild.

Answer 2

Maybe the link that you are missing is this understanding of the word 'distribution', as used in mathematical analysis.

A (complex-valued) distribution, sometimes also called 'generalized function', is a function $$\varphi: \mathcal S \to \mathbb C, \tag1$$ where $\mathcal S$ is a space of 'well-behaved' functions known as test functions. The precise choice depends on what you're doing, but it is typically something like the set of smooth functions with compact support.

The reason that we call them 'generalized functions' is that if you have a function $f:\mathbb R\to\mathbb C$, you can normally also define a distribution $\varphi_f$, whose value on a test function $g\in\mathcal S$ is given by $$ \varphi_f(g) = \int_{-\infty}^\infty f(x)^*g(x)\mathrm dx. $$ However, this class can also include pathological objects like the Dirac delta, which fit the mold of $(1)$ but for which the use of the language of normal functions can be quite awkward.

In any case: an operator-valued distribution is a linear function that, like in $(1)$ accepts test functions in $\mathcal S$, but which returns operators that act on your Hilbert space $\mathcal H$ of interest.

I will leave the rigorous details to others, but hopefully this will be useful for you, if this was indeed the gap that needed to be filled, and, if not, then for others after you ;-).