This is a follow up to my question The Sobolev Space $H^{1/2}(∂Ω)$ as the Quotient Space $H_1(Ω)/\ker(\text{tr})$. Whilst this is another question related to the same space, I believe it focuses on another aspect of the space, and thus merited another question.
Consider the Sobolev space $$H^{1/2}(\partial Ω) = \{ u ∈ L^2(\partial Ω) \;|\; ∃ \tilde u ∈ H^1(Ω)\colon u = \text{tr}(\tilde u) \}$$ together with the norm $$\| u \|_{H^{1/2}(\partial Ω)} = \inf \{ \| \tilde u \|_{H^1(Ω)} \;|\; \text{tr}(\tilde u) = u\}.$$ On the one hand, this space is recognisable as the range of the trace operator, $\text{tr}$, but on the other hand, this space is often constructed as the quotient of $H^1(\Omega)$ by $\ker(\text{tr})$.
I have read that $H^{1/2}(∂Ω)$ is isomorphic to the quotient $H_1(Ω)/\ker(\text{tr})$. I have also read that $H^{1/2}(∂Ω)$ is unitarily equivalent to $\left(\ker(\text{tr})\right)^\bot$. Questions: how is it that these two representations of $H^{1/2}(∂Ω)$ are related? Are they effectively the same space? Is there some difference between them?
An important subset of $H^1(\Omega)$ is $H^1_0(\Omega)$, the Sobolev space of those functions in the domain of the gradient with vanishing trace (those functions who vanish at the boundary, $\partial\Omega$). It seems to me that $H^1_0(\Omega)$ should be intimately related to $H^{1/2}(∂Ω)$, in particular when regarded as being unitarily equivalent to $\left(\ker(\text{tr})\right)^\bot$. Questions: concretely, what is the connection between these two spaces (in particular, when describing functions behaviour at the boundary)? Am I correct in saying that $\left(\ker(\text{tr})\right)^\bot$ contains precisely those functions who do not vanish at the boundary (or, those with non-zero trace)?
