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
Have you come across the 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 .