Page 311 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Measure Theory and Functional Analysis

Notes 28.3 Keywords

Positive Operator: A self adjoint operator on H is said to be positive if A 0 in the order
relation. That is

if Ax,x 0 x H.
Self Adjoint: An operator T on a Hilbert space H is said to be self adjoint if T*=T.

28.4 Review Questions

1. Define a new operation of “Multiplication” for self-adjoint operators by
A A A A
A  A 2 1 2 2 1 , and note that A  A is always self-adjoint and that it equals A A 2
1
1
2 1 2
whenever A and A commute. Show that this operation has the following properties:
1 2
A  A A  A ,
1 2 2 1
A  A A A  A A  A ,
1 2 3 1 2 1 3
A  A A  A A  A ,
1 2 1 2 1 2
and A I I A A.  Show that

A  A  A A  A  A 3 whenever A and A commute.
1 2 3 1 2 1 3
2
2. If T is any operator on H, it is clear that Tx,x Tx x T x ; so if H 0 ,we have
2
sup Tx,x / x : x 0 T . Prove that if T is self-adjoint, then equality holds here.

28.5 Further Readings

Books Akhiezer, N.I.; Glazman, I.M. (1981), Theory of Linear Operators in Hilbert Space

Yosida, K., Functional Analysis, Academic Press

Online links www.ams.org/bookstore/pspdf/smfams-14-prev.pdf-UnitedStatesmath
world.wolfram.com

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