Extensions of Projection-valued Measures
If π is an additive projection-valued measure on (X, M), then the map
extends to a linear map on the vector space of step functions on X. In fact, it is easy to check that this map is a ring homomorphism. In fact this map extends in a canonical way to all bounded complex-valued Borel functions on X.
Theorem. For any bounded M-measurable function f on X, there is a unique bounded linear operator Tπ(f) such that
for all ξ, η ∈ H. The map
is a homomorphism of rings.
Read more about this topic: Projection-valued Measure
Famous quotes containing the words extensions of, extensions and/or measures:
“If we focus exclusively on teaching our children to read, write, spell, and count in their first years of life, we turn our homes into extensions of school and turn bringing up a child into an exercise in curriculum development. We should be parents first and teachers of academic skills second.”
—Neil Kurshan (20th century)
“If we focus exclusively on teaching our children to read, write, spell, and count in their first years of life, we turn our homes into extensions of school and turn bringing up a child into an exercise in curriculum development. We should be parents first and teachers of academic skills second.”
—Neil Kurshan (20th century)
“thou mayst know,
That flesh is but the glass, which holds the dust
That measures all our time;”
—George Herbert (1593–1633)