I am quite confused about linear operators that are defined as acting on a Hilbert space $\mathcal{H}$ and their representations. The operators form an operator algebra and as such can be represented in a representation space $V$ just like any algebra $\mathcal{A}$ via a representation $\rho: \mathcal{A}\rightarrow End(V)$. If I chose as representation space the very Hilbert space $\mathcal{H}$ the operators are defined on, are the “abstract” operator $A \in \mathcal{A}$ and its representation $\rho(A) \in End(\mathcal{H})$ not one and the same? Is there any difference?
2026-02-22 23:14:41.1771802081
Difference between operator defined on a space and operator represented in a space
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Yes they are the same, and no, there is no difference.
When an algebra $\mathcal{A}$ is defined as a subalgebra of $\mathrm{End}(\mathcal{H})$, then the natural representation $$\rho:\mathcal{A}\to\mathrm{End}(\mathcal{H})$$ is just the embedding map. In this case, the elements $A\in\mathcal{A}$ aren't really "abstract" operators, they are defined as elements $A\in\mathrm{End}(\mathcal{H})$.