Exercise 40, § 5

Let E be a complex Hilbert space.

a) Let $u$ be a normal partial operator on E and let $(j,|u|)$ be its polar decomposition. One has $|u^*|=|u|$ and the polar decomposition of $u^*$ is $(j^*,|u|)$. One has $j|u|=|u|j$.

b) Let $u$ be a self-adjoint partial operator on E and let $(j,|u|)$ be its polar decomposition. One has $j=$ s($u$), where the function s is defined by

1 if $x >0$

s($x$) $=$ 0 if $x= 0$

$-1$ if $x <0$.

c) Let $u$ be a self-adjoint partial operator on E and let $(j,|u|)$ be its polar decomposition. Let D denote the domain of $|u|^{1/2}$, and define a partial Hermitian form $q$ with domain D by putting

The partial operator representing $q$ coincides with $u$; if $u$ is positive, then $q$ is the positive partial form associated with $u$.

d) Let $u$ and $v$ be self-adjoint partial operators on E. Let $q_u$ and $q_v$ denote the partial Hermitian forms defined as in question c). Suppose that dom($q_u$)$\cap$ dom($q_v$) is dense in E. The sum of $u$ and $v$ in the sense of Hermitian forms is the partial operator $u$ + $v$ on E representing the partial form

with domain dom($q_u$)$\cap$ dom($q_v$). Now prove that the partial operator $u$+ $v$ is an extension of $u+v$; if $u$ and $v$ are positive, then $u$+ $v$ is a positive self-adjoint partial operator.

e) Give an example where the domain of $u+v$ is not dense, but $u$ + $v$ is defined.

f ) Let $n\geqslant 1$ be an integer and $S\subset \mathbf{R}^n$ a closed set of Lebesgue measure zero. Let $v$ be a positive measurable function on $\mathbf{R}^n$ for the Lebesgue measure such that $v$ is locally integrable on $\mathbf{R}^n-$ S. The self-adjoint partial operator Δ + $m_v$ is defined.