By Peter Fillmore

Those notes characterize the substance of a process lec-

tures I brought at Indiana collage in 1967. The viewers

was accustomed to the elemental concept of Hilbert areas and op-

erators up during the spectral theorem and the idea of

spectral multiplicity. for plenty of attention-grabbing and useful 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 toes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

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And trans . ] (Jerusalem: American Academy for Jewish Research, 1966) : 23. An English translation of the introduction to this commentary was published by Moshe Sokolow, "Saadiah Caori's Prolegomenon to Psalms", PAAJR , 51 (1984) : 131-74. 6 Ibid . 4. R. Saadiah's theory of knowledge has been explored in studies by Israel Efros and Abraham Heschel in JQR, 33 (1942): 133-70,265-313. 7 Ibid. 4 (Kafih, 13; Rosenblatt, 15) . 8 Ibid. 5. 30 CHAPTER ONE: R. SAADIAH GAON In light of his theological convictions, R.

Even those who wrote philosophic treatises and went on to write commentaries, Gersonides being a good example of this phenomenon, appear to alter their manner of thinking as they move from one genre to the next. I will make reference to the commentaries of the thinkers whose works are covered in this study to the extent I feel they are important for understanding certain developments in their thought. 12 The broader cultural (as well as social) context in which the treatises were written is not one upon which I dwell in the present study.

This issue too is su bordinate to o ther concerns, primarily to th e problem of God's esse nce, and secon darily to the probl em of the divine law. Most of hi s remarks on the nature of prophecy occur in th e section on the un ity of Go d . They are formulated to meet so me of the challe nges to h is co nceptual app ro ach . Overall , the p roblem of Nahum Glatzer ed . (Philadelphia: J ewish Pu bli catio n Society, 1974) : 246 68 . THE BOOK OF B ELIEFS A N D O PINI ONS 33 "what" exactly is prophecy is of no gr eater concern to him in th e tr eatise than the problems of "why" is there prophecy, "h ow" is it verified, the proper understanding of th e prophetic m essages , an d th eir r elation to th e co ncl us ions based upon r ati onal speculation.