Hello, could you please help me to solve this problem:
Let - the algebra of bounded linear operators on a Hilbert space , and is positive. Then sequence is monotone increasing to the range projection of
functional calculus for selfadjoint operators? If so, you can translate this problem into a problem about functions on the spectrum of the operator . In fact, you just need to show that the sequence of functions is monotone increasing (on the set ) to the function , where and for
To do this without the functional calculus would be harder. You could start by observing that the operators , and all commute with each other. Let . If n>m then . The right side of that is a product of three commuting positive operators and is therefore positive. That shows that the sequence is monotone increasing.
To complete the proof, you need to know that . This again is easy if you know a bit of spectral theory. In fact, it follows from the fact that on .
First, notice that . It follows that if is in the range of (where ) then as . Thus strongly on the range of and hence (since for all n) on the closure of the range of .
The orthogonal complement of the range of is the null space of . So if is in that orthogonal complement then . Thus strongly on the orthogonal complement of the range of .
Put that all together to see that converges strongly to the range projection of .