SEMINORMAL OPERATORS AND WEYL SPECTRA

Nihonkai Math J

Vol 8 '1991:. 77-83

S E M I N O R M A L OPERATORS A N D WEYL SPECTRA

A r 3 s , r r i A c T . In tliis paper w e s h o w t h a t the WeyI s p e c t r u m of a seniinorrual operator 7' satisfies t h e spectral mapping thcoreiri for a n y analytic futiction f on a neighborhood of n ( 2 ' ) a n d Weyl's theorem holds for f ( T ) . Finally

we give conditions for a n operator t o be of t h e form unitary

+

compact. and answer an old question of O b e r a i .

0. Introduction. Throughout this paper let.

H

denote an infinite di- iiierisiorial Hilbert space and

B( H )

the set of all bounded linear operators on

H .

If

T E B ( H ) ,

w e write a j T ) for the spectrum of

T .

;7o(T) for. the set of eige1ii.aliir.s o f

T ,

.irof(T) for the set of eigenvalues of finite niultiplicity, a i d

.UU(

T )

for the isolatcd points of a (

T )

tiiat arc eigciivalues of finite multiplic- ity. If

E

i s a subset of @, we write iso

E

for the w t of isolated points of

E .

-4ii opcrator

T

E

B( H )

is said to be FredholTn if its range ran

T

is closed and both the null spaces ker

T

and ker

T *

are fiiiit,c diniensional. The i n d e x o f a Fredholm operator

T ,

denoted liy

i(

T ) .

is defined liy

i ( T )

= dirriker

T

- dim kcr

T * .

The e s s e n t i a l s p e c t r u m of

T ,

denoted by a , ( T ) , is defined by

a , ( T ) =

{

X

E

@ :

T

- X I is riot Fredholm}.

,4

Fredholm operator of index zero is called a W e y l operator. spectru7n of

T ,

denoted by

d ( T ) ,

is defined by

The Weyl

It was shown ([2]) that for any operator

T , a , ( T )

c

d ( T )

c

a ( T ) ,

and

; ( T )

is a nonempty compact subset of @.

Recall ([9], [12]) that an operator

T

E

B ( H )

is said to be s e m i n o r m a l if

either

T

or

T *

is hyponormal. Every hyponormal operator is seminormal,

K e y words a n d p h r a s e s . Weyl, d o m i n a n t , seminormal, polynomially compact

1991 Mathematics S u b j e c t Classification. 4 7 A 1 0 , 4 7 A 5 3 , 4 7 8 2 0 .