Appendix III

APPENDIX I11

1.

Support of an operator.

L e t /-/ be a complex H i l b e r t space, and T E L ( f f ) . The s e t of = 0 i s a closed l i n e a r subspace of ff. Let be i t s orthogonal complement. The p r o j e c t i o n E = P x i s c a l l e d For every y E ff, we have y - Ey E XI, hence t h e support of T . T ( I - E)Y = 0 . Consequently, T = TE. Conversely, l e t E l be a p r o j e c t i o n of L(ff) such t h a t TE1 = T ; w e have, f o r every a E ff, T ( I - E 1 ) z = 0 , hence ( I - E~)ZEX'; hence I - E l S I - E , E12E. Thus, E i s t h e s m a l l e s t of t h e p r o j e c t i o n s El of L(ff) such t h a t

x

x ~ fsuch f t h a t Tx

TE1 = T.

The c l o s u r e of T ( H ) i s a closed l i n e a r subspace Y of ff. Let F = Py. I t i s c l e a r t h a t F i s t h e s m a l l e s t of t h e p r o j e c t i o n s F 1 of L(ff) such t h a t F I T = T o r , which comes t o t h e same t h i n g , T*Fl = T*. Hence F i s t h e support of T*.

Partial isometries.

2. Let U

E

L ( f f ) , and

W e say t h a t U i s a p a r t i a l E i t s support. = E(ff). Then, U(ff) = U(x) i s a

isometry i f U i s i s o m e t r i c on

x

closed l i n e a r subspace Y of ff, and U maps x i s o m e t r i c a l l y onto Y. L e t F = Py. We say t h a t E ( r e s p . x) i s t h e i n i t i a l project i o n ( r e s p . t h e i n i t i a l subspace) of U, and t h a t F ( r e s p . Y) i s t h e f i n a l projection ( r e s p . t h e f i n a l subspace) of U. Let

XEX, y

= UXE

(xlz) = (xlEz)

=

Y.

For every a ~ f f ,we have

(UxlUEz) = (ylUz),

hence

x

=

U*y.

x

Thus, t h e mapping x + U x of onto Y has f o r i t s i n v e r s e ( i s o m e t r i c ) mapping t h e mapping y -+ U*y of Y onto x. Since, furthermore, t h e support of U* i s F, we see t h a t U* i s a p a r t i a l isometry, with i n i t i a l p r o j e c t i o n F, and f i n a l p r o j e c t i o n E . We a l s o see t h a t U*U = E, UU* = F. Conversely, l e t V E L(ff) be such t h a t V*V i s a p r o j e c t i o n G . Then, f o r every x E ff, we have

366

hence V i s i s o m e t r i c on G ( f f ) and z e r o on G ( f f ) l , which proves t h a t V i s a p a r t i a l i s o m e t r y . S i m i l a r l y , i f W E L(ff) and i f WW* i s a p r o j e c t i o n , W* i s a p a r t i a l i s o m e t r y , and hence W i s a p a r t i a l isometry.

Polar decomposition of an operator.

3.

Let T E

L(H),

and Y = F ( f f ) .

F

E t h e s u p p o r t of T , t h e s u p p o r t of T*, X = E ( f f ) W e put: I T ! = (T*T)'. W e have, f o r e v e r y x c f f ,

(1

Tx

(I2

= (T*Tx(x) =

(1

( T ( x(I.

Hence I T ( h a s s u p p o r t E and, c o n s e q u e n t l y , = x. Furthermore, t h e mapping I T 1 5 + TZ i s a l i n e a r isometry of IT1 (H) o n t o T ( f f ) , and t h e r e f o r e e x t e n d s t o a l i n e a r isometry V of x o n t o Y. L e t U be t h e p a r t i a l isometry w i t h s u p p o r t E which c o i n c i d e s w i t h V on x; t h i s p a r t i a l isometry h a s E a s i n i t i a l p r o j e c t i o n , and F as f i n a l p r o j e c t i o n . W e have T = U I T I , an e q u a l i t y c a l l e d t h e polar decomposition of T. On t h e o t h e r hand, i f w e have an e u q a l i t y T = U 1 T 1 , where T 1 i s p o s i t i v e h e r m i t i a n and where U1 i s a p a r t i a l isometry whose i n i t i a l p r o j e c t i o n i s t h e s u p p o r t of 2 hence T1 = I T ( , and t h e n T l , t h e n w e have T*T = T I U f U I T l = T1,

u1

=

u.

W e have T* = ITIU* = U*(UITIU*);

t h e o p e r a t o r UITIU* i s p o s i t i v e h e r m i t i a n , w i t h s u p p o r t F; and U* i s a p a r t i a l i s o m e t r y , w i t h i n i t i a l p r o j e c t i o n F , and f i n a l projection E. Hence t h e e u q a l i t y T* = U*(U(TIU*) i s t h e p o l a r decomposition of T*. Thus, IT*\ = UITIU*,

T

= U*(T*IU

Reference : J . VON NEUMANN, Uber adjungierte FunktionaZoperatoren (Ann. Math., 33, 1932, 294-310).