Notes on Operator Theory

These notes represent the substance of a course of lec-
tures I delivered at Indiana University in 1967. The audience
was familiar with the basic theory of Hilbert spaces and op-
erators up through the spectral theorem and the theory of
spectral multiplicity. For many interesting and helpful con-
versations I am indebted to R. G. Douglas, and to my col-
leagues Arlen Brown, J. G. Stampfli, D. M. Topping and
J..P. Williams.
Contents:
I n trod u cti on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
o. Hyponormal Oper ators .... . . . . . . . . . . . . . . . . . 7
1. Shi fts ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 . Mod e 1 s . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 21
3. Invariant Subs paces of Simple Shifts........ 25
4. Invariant Subspaces of General Shi fts ....... 31
5. Analytic Representation of the Unilateral
Shi ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6. Contractions..........
. . . . . . . . . . . . . . . . . . 49
7. Dilations ............................... 57
8. Naimark's Theorems on Dilations........... 65
9. Contractive Semi groups ................... 77
10 H . . S b 95
.. Y pe rl n v ar I an t u spa c e s .................. .
11. Invariant Subspaces for Compact Operators. .. 107
R e fer en c e s .............................. 115
Index ................................... 121