Abstract:

A classification of partial isometries defined on a Hilbert space can be made in terms of positions that two subspaces, the initial space and the final space, form. When the orthogonal projections onto two subspaces commute, any power of the partial isometry remains also a partial isometry. This type of partial isometry is called a power partial isometry. Halmos and Wallen proved that every power partial isometry is a direct sum of unitary operators, isometries, co-isometries, and truncated shifts. Another special type of partial isometries, a generic partial isometry, arises when the two subspaces are in generic position. The numerical range of an operator A on a Hilbert space H is defined as W (A) = {⟨Aξ, ξ⟩ : ξ ∈ H, ∥ξ∥ = 1}. An elaborate description of the numerical range of partial isometries will be given based on the classification of partial isometries.