Let $X \equiv \mathbb{R}^\mathbb{N}$ denote the space of real sequences with the product topology.
Suppose that $T \colon \mathcal{H} \to X$ is a linear operator, where $\mathcal{H}$ is a real, separable Hilbert space.
In my application, $T$ is an operator which actually has the property that the range of $T$ is included in $\ell^2(\mathbb{N})$; moreover as a map to $\ell^2$, it is bounded. In other words, $T = \iota \circ S$ where $S \colon \mathcal{H} \to \ell^2$, and $S$ is a bounded linear operator and $\iota \colon \ell^2 \to X$ is the inclusion map.
I want to consider $T$ mapping to $X$, since I need to consider quantities such as $y + Tx$ where $y \in X, x \in \mathcal{H}$, so I really do want to consider the included map $T$.
I have the following questions:
- Is there a reasonable way to define an adjoint of $T$? For instance, I was thinking that one could some how extend $S^\ast$. However, I am not sure how to define the values of $T^\ast$ when $x \in X \setminus \ell^2$.
- Is it possible to define a notion of singular value decomposition for this operator? Again, here I can imagine that one writes $S = U \Sigma V^\ast$, where $U, V \colon \ell^2 \to \mathcal{H}$ are unitary and $\Sigma$ is a positive, diagonal operator w.r.t. standard basis. Thus $T = \tilde U \Sigma V^\ast$, where $\tilde U = \iota \circ U$. However, related to the previous question, I don't know what $\tilde U^\ast$ means in this context, much less if something like $\tilde U^\ast \tilde U$ makes sense.
The main challenges to the above questions seem to be that $\mathbb{R}^\infty$ is non-normable and has no inner product. Any references that treat this type of question would also be quite useful.
I am skeptical about your second question, but there is a simple answer to your first question. For every bounded linear operator $T$ between topological vector spaces $E$ and $F$, there is an adjoint operator $T^\ast$ between $F^\ast$ and $E^\ast$ given by $T^\ast \phi=\phi\circ T$. In other words, $T^\ast$ is just the restriction of the algebraic adjoint to the topological duals.
Of course, in general the topological dual of a topological vector space can be trivial, in which case this adjoint is not an interesting object, but for the special case of $\mathbb R^{\mathbb N}$ there are plenty of bounded linear functionals. Moreover, if $E$ and $F$ are (real) Hilbert spaces, then this notion of adjoint coincides with the usual one up to the identification $E\cong E^\ast$ via the Riesz representation theorem.
In your special case, the topological dual of $X$ is $c_c$, the space of sequences with finitely many non-zero entries, and the adjoint of $T$ is the restriction of the adjoint of $S$ to $c_c$.