By Peter Fillmore

Those notes signify the substance of a process lec-

tures I introduced at Indiana college in 1967. The viewers

was conversant in the elemental idea of Hilbert areas and op-

erators up during the spectral theorem and the idea of

spectral multiplicity. for plenty of attention-grabbing and necessary con-

versations i'm 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 easy Shifts........ 25

4. Invariant Subspaces of normal Shi fts ....... 31

5. Analytic illustration of the Unilateral

Shi feet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . forty five

6. Contractions..........

. . . . . . . . . . . . . . . . . . forty nine

7. Dilations ............................... fifty seven

8. Naimark's Theorems on Dilations........... sixty five

9. Contractive Semi teams ................... seventy seven

10 H . . S b ninety five

.. 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 .............................. a hundred and fifteen

Index ................................... 121