[Math] Find the adjoint of the right shift operator in $\ell^1$.

adjoint-operatorsfunctional-analysisreal-analysis

Find the adjoint of the right shift operator $T$ in $\ell^1$.

More specifically, $T: X \longrightarrow Y$ defined by

$$Tx= T(x_1, x_2, \dots, x_n, \dots) = (0, x_1, x_2, \dots, x_n\dots) = y$$

Attempt

I fist showed $T$ was linear, then bounded by noticing sum of the series generated by $(|x_n|)$ is the same as the series generated by $(|y_n|)$. Now it remains for me to find the adjoint operator of $T$.

I'm sort of confused at this step though because I think I'm supposed to be finding the adjoint in the sense of $T^\times: Y' \longrightarrow X'$ defined by (in Kreyszig's functional analysis book)

$$f(x) = \left(T^\times g\right)(x) = g(Tx), \text{where } g \in Y'$$

which I don't believe is the same as the Hilbert adjoint (which my brain keeps circling back to). Do I need to find $f$ explicitly here?

More or less I'm hoping someone can clearly explain what the idea is and maybe give me a step in the right direction. Thank you.

Best Answer

Let me use a different notation, which really looks like the one used in the Hilbert space setting, i.e. let $\langle g, y \rangle $ denote $g(y)$ the pairing of $g \in Y^\prime$ and $y\in Y$. The adjunct operator of $T:X\to Y$ is then defined as the unique $T^\times :Y^\prime \to X^\prime$ s.t.:

$$\langle T^\times g, x \rangle = \langle g, Tx \rangle \; .$$

I used the angular brackets $\langle \cdot ,\cdot \rangle$ because of two well-known features of $\ell^p$ spaces, namely:

  1. for each $(x_n) \in \ell^p$ and $(f_n) \in \ell^{p^\prime}$ ($p^\prime$ is the conjugate exponent of $p$) one has $(f_nx_n) \in \ell^1$;
  2. each functional $f \in (\ell^p)^\prime$ can be represented by a unique sequence $f=(f_n)\in \ell^{p^\prime}$ s.t. equality: $$f(x) = \sum_{n=1}^\infty f_nx_n =\langle f , x\rangle$$ holds for all $x=(x_n) \in \ell^p$.

Usually such features are summarized in the frase:

$\ell^{p^\prime}$ is the dual of $\ell^p$ and the pairing coincides with the dot-product.

Therefore your problem recasts in a (probably) more familiar form: for each fixed $g=(g_n) \in \ell^\infty = \ell^{1^\prime}$, find the sequence $T^\times g=:f=(f_n) \in \ell^\infty$ s.t.: $$\langle f, x \rangle = \langle g, Tx \rangle $$ for all $x=(x_n) \in \ell^\infty$.

This is exactly what you do when you have to "find" the adjunct operator in the Hilbert space setting, isn't it? ;-)

Related Question