Page 300 - DMTH505_MEASURE_THEOREY_AND_FUNCTIONAL_ANALYSIS

Unit 27: The Adjoint of an Operator

 The adjoint operation T T* on H has the following properties: Notes

(i) T 1 T * T * 1 T * 2
2
(ii) T T * T * T * 1
1
2
2
(iii) T * T *

(iv) T * T

(v) T * T T 2

27.3 Keywords

Adjoint of the Operator T: Let T be an operator on Hilbert space H. Then there exists a unique
operator T* on H such that

(Tx,y)= (x,T*y) for all x, y H
The operator T* is called the adjoint of the operator T.
Conjugate of the Operator T on H: T gives rise to an unique operator T* and H* such that (T*f) (x)
= f(Tx) f H * and x H. The operator T* on H* is called the conjugate of the operator T on H.

27.4 Review Questions

1. Show that the adjoint operation is one-to-one onto as a mapping of H into itself.
2
2. Show that TT * T .
3. Show that O*=O and I*=I. Use the latter to show that if T is non-singular, then T* is also

1 1
non-singular, and that in this case T * T *.
27.5 Further Readings

Books N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Space, Vol. II,
Pitman, 1981.
K. Yosida, Functional Analysis, Academic Press, 1965.

Online links www.math.osu.edu/ gerlach.1/math.BVtypset/node 78.html.
sepwww.standford.edu/sep/prof/pvi/conj/paper_html/node10.html.

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