More specifically, finding self-adjoint extensions, with various requirements, of symmetric operators is equivalent to finding unitary extensions of suitable partial isometries.
12.
The Toeplitz structure of " A " means that a " truncated " shift is a partial isometry on \ mathcal { H }.
13.
In that case, the solutions to the problem are in bijective correspondence with minimal unitary extensions of the partial isometry " V ".
14.
In that case, the operator is a partial isometry, that is, a unitary operator from the range of " T " to itself.
15.
A similar but weaker statement holds for the partial isometry : " U " is in the von Neumann algebra generated by " A ".
16.
A similar but weaker statement holds for the partial isometry : the polar part " U " is in the von Neumann algebra generated by " A ".
17.
T is the partial isometry that vanishes on the orthogonal complement of " U " and " A " is the isometry that embeds " U " into the underlying vector space.
18.
This is because that in the finite dimensional case, the partial isometry " U " in the polar decomposition " A " = " UP " can be taken to be unitary.
19.
When " B " 1 and " B " 2 are not assumed to be minimal, the same calculation shows that above claim holds verbatim with " U " being a partial isometry.
