# Transpose of an operator acting in a complexifing Hilbert space

The context of my question is the following.

Up to my knowledge when we are working in a complex Hilbert space $$\mathcal{H}$$ there exists the notion of adjoint for densily defined operator $$A:\mathcal{D}\subseteq \mathcal{H}\to \mathcal{H}$$ and it is noted $$A^{\ast}$$, defined by the property $$(Au,v)=(u,A^{\ast}v)$$ for all $$u\in \mathcal{D}$$ and $$v$$ also in a dense.

But define the traspose $$A^{T}$$ of an operator $$A$$ it is not even abailable uless we work in an specific basis of $$\mathcal{H}$$, suposse $$u_{n}$$ is such a basis, then the transpose will be definded by $$(Au_{i},u_{j})=(A^{T}u_{j},u_{i})$$.

But I'm currently working on a specific Hilbert space $$\mathcal{H}=\mathcal{F}_{\mathbb{C}}$$ which is the complexification a a real Hilbert space $$\mathcal{F}$$. Clearly the interior product in $$\mathcal{F}$$ which I call $$(,)_{\mathbb{R}}$$ is extended to an internal product in $$\mathcal{H}$$ which I call $$(,)_{\mathbb{C}}$$, so for operators in $$B(\mathcal{H})\simeq B(\mathcal{F})_{\mathbb{C}}$$ we have a notion of adjoint, becausse we have an internal product. But now thanks to the structure of complexification we have also a notion of conjuate for vectors and for operators in $$B(\mathcal{H})$$.

After a lot of computations about this thing I arrive to the idea that in this context one can define the notion of transpose by $$A^{T}=\overline{A^{\ast}}$$.

Here some of my questions...

1. ¿Is there analogous of this definition in some textbook? Or ¿do you think it is fine?

2. I found that, conjugate and adjoit conmute in $$B(\mathcal{H})$$, i.e. $$\overline{(A^{\ast})}=( \overline{A} )^{\ast}$$ ¿is it right?

3. I also found that $$\overline{(u,v)_{\mathbb{C}}}=(\overline{u},\overline{v})_{\mathbb{C}}$$, for $$u,v\in \mathcal{H}$$, again the same ¿Is it right?

I hope I were clear. Most of the ideas were partially inspired by this text where you can find the definition of $$(,)_{\mathbb{C}}$$ in terms of $$(,)_{\mathbb{R}}$$ and related things.

• I believe $\mathcal F$ is called a real form of $\mathcal H$. A real form is a real subspace of $\mathcal H$ so that $(v,w)\in\Bbb R$ for all $v,w\in\mathcal F$ and $\mathcal F+i\mathcal F=\mathcal H$. The data of a real form is equivalent to an anti-unitary involution (the complex conjugation) on $\mathcal H$, which I think is called a real structure. Jul 16, 2020 at 10:23