A Theory of Cross-Spaces. (AM-26), Volume 26 9781400881963

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ANNALS OF MATHEMATICS STUDIES Number 26

ANNALS OF MATHEMATICS STUDIES

Edited by Emil Artin and Marston Morse 7. Finite Dimensional Vector Spaces, by 11.

Paul

Introduction to Nonlinear Mechanics, by N.

14. Lectures on Differential Equations, by

R.

H alm os

and N.

Krylo ff

B o g o l iu b o f f

Solom o n L e fsc h etz

15. Topological Methods in the Theory of Functions of a Complex Variable, by M a r s t o n M o r s e 16. Transcendental Numbers, by

C a r l L u d w ig S i e g e l

17. Probleme General de la Stabilite du Mouvement, by M. A. 18. A Unified Theory of Special Functions, by C. A. 19. Fourier Transforms, by S.

B och n er

and

L ia p o u n o f f

T r u esd ell

K . C h a n d r a sek h a ra n

20. Contributions to the Theory of Nonlinear Oscillations, edited by S. L e f s c h e t z 21. Functional Operators, Vol. I, by

J ohn von Neum a n n

22. Functional Operators, Vol. II, by

J ohn von N eu m a n n

23.

Existence Theorems in Partial Differential Equations, by

D oroth y

B e r n s t e in

24. Contributions to the Theory of Games, edited by A. W. 25. Contributions to Fourier Analysis, by A. Z y g m u n d , W. A. P. C a l d e r o n , and S. B o c h n e r 26. A Theory of Cross-Spaces, by

T u cker

T ra n su e,

M.

M o r se,

R o bert Sch atten

27. Isoperimetric Inequalities in Mathematical Physics, by G. G. S z e c o

Polya

and

A THEORY OF CROSS-SPACES BY ROBERT SCHATTEN

PRINCETON P R I N C E T O N U N I V E R S I T Y PRES S

195 °

Copyright © 1950, by P r i n c e t o n

U n iv e r s ity P re ss

Printed in the United States of America by W e s t v ie w P r e s s , Boulder, Colorado P r in c e t o n U n iv e r s it y P r e s s O n D em a n d E d it io n ,

1985

TA BLE OF CONTENTS Page INTRODUCTION 1. S ta te m e n t of the p ro b le m 2. P u rp o se of th is exp o sitio n 3 • A cknow ledgem ent 4. P la n of study 5. O utline of r e s u lts

I 3 6 6 8

NOTATIONS AND CONVENTIONS I.

H.

16

TH E A L G E B R A O F E X P R E S S IO N S 1. T he e x p r e s s io n s ETS-.f*.® g t 2. The lin e a r s p a c e s and 3. T ra n s fo rm a tio n s on e x p r e s s io n s CROSSNORM S 1. The norm ed lin e a r s p a c e s 2. C r o s s n o rm s 3. The bound a s a c r o s s n o r m 4. The g r e a te s t c r o s s n o r m

8t

T>,©ai7^a. an [16] quoted in the c h ro n o lo g ic a l o rd e r of th e ir a p p ea ran ce, y e t, the p re s e n t tre a tm e n t is d e s ira b le fo r the follow ing re a s o n :

The seem in g ly in n ocen t

- - a t the s t a r t — p ro b le m of “c r o s s in g " two B a n a c h s p a c e s , grad ually grew into an e x te n siv e field w ith in te re s tin g a p p lic a tio n s . A cco rd in g ly , d efin itio n s, s ta te ­ m e n ts and n otatio n (w hich p re s e n ts quite a p ro b le m in its e lf) had to be re v ise d fro m tim e to tim e to su it the new n e ed s.

The p r e s e n t e xp o sitio n a ls o inclu d es

m o st of th e s e s c a tte r e d r e s u lts in a unified th e o r y . A few r e s u lts h e r e in , have not b een published b e fo r e . W h e re v e r p o s s ib le , the p u blish ed ones have b een re fin e d . W hile the p r e s e n t th e o ry — w hich h a s tu rn ed out to fu rn ish an e ffe c tiv e to o l in d ealing w ith B a n a c h sp a c e s w hose e le m e n ts a r e o p e ra to rs on som e B a n a ch s p a c e — is in an advanced s ta g e , it is fa r fro m being c o m p le te . A s a m a tte r of f a c t a nu m ber of in te re s tin g p ro b le m s a r e s t i ll open. Som e a r e m entioned in the body of th is p&per, and the m a in d ifficu lty in th e ir solution is pointed ou t. It i s hoped th a t th is p re s e n ta tio n w ill induce the in te re s te d r e a d e r to fu rth e r in v e stig a tio n s in th is p ro m isin g fie ld .

6

INTRODUCTION 3. A cknow ledgem ent. A t th is point it se e m s p ro p er to acknow ledge th at the a u th o r's d is c u ss io n s

w ith P r o fe s s o r J . von Neumann in 1 9 4 4 -4 6 (during the a u th o r's m e m b ersh ip at The In stitu te fo r Advanced Study) follow ed up by an exchange of c o rre sp o n d e n c e in 1 9 4 6 -4 8 have played a d e c is iv e p a rt in p re p a rin g the foundation fo r th is d ra ft. N e ed less to say th a t the id ea s contained in th is d ra ft w ere o rig in a ted by both P r o f e s s o r von Neumann and the a u th o r. Som e of th e s e , w ere published (although in d iffe re n t fo rm ) in th e ir jo in t p a p e rs f l 5 j and [ l 6 j . W hile the au th o rs a ssu m e s fu ll r e s p o n s ib ility fo r the sh o rtco m in g s th a t m ay be contained in th is e x p o sitio n , the m e r its and c r e d its it m ay have m u st be sh ared with P r o fe s s o r von N eumann.

4 . P la n of study. To have som e id ea what to e x p e ct, suppose f i r s t that ^ and ^?^are fin ite , say p and q d im en sion al B a n a ch s p a c e s , w hose e le m e n ts w ill be denoted by y

f and g , w h ile denote by F

and G

-ft

and

denote th e ir con ju g ate s p a c e s whose e le m e n ts we

.

A s w as m entioned b e fo r e , we m ay then in te rp re t f R g fo r in s ta n c e , a s the o p e ra to r

F (f)g fro m ^

into

(o r G (g)f f r o m l ^ i n t o l ^ ) of ran k ^

1.

T he e x p r e s s io n ! § L ^ t fj® g