Int. Journal of Math. Analysis, Vol. 6, 2012, no. 31, S. Panayappan

Transcription

1 It Joural of Math Aalysis, Vol 6, 0, o 3, O Power Class ( Operators S Paayappa Departet of Matheatics Goveret Arts College, Coibatore 6408 ailadu, Idia paayappa@gailco N Sivaai Departet of Matheatics ailadu College of Egieerig,Coibatore ailadu, Idia sivaaitce@gailco Abstract I this paper we itroduce the ew class power class( Hilbert space H A operator L(H) is powerclass( operators actig o a if ( ) * We ivestigate soe basic properties of such operator I geeral a power class Q operator eed ot be a oral operator ( ) Matheatics Subject Classificatio: 47B0, 47B99,47B5 Keywords: Noral, -Noral, power quasi oral, class (, Hilbert space Itroductio hroughout this paper H is a Hilbert space ad L (H ) is the algebra of all bouded liear operators actig o H A operator L(H ) class if * ( ), is called oral if is called (, is -oral if,

2 54 S Paayappa ad N Sivaai is power quasi oral if ( ) ( ) ( ) ( ) ad is quasi oral if Mai Results I this sectio we ivestigate soe properties of operators i power class( heore If powerclass( the so are (i) k for ay real uber k (ii) ay S L(H ) that is uitarily equivalet to (iii) the restrictio of to ay closed subspace M of H that reduces M Proof (i) he proof is straightforward (ii) Let S L(H ) be uitarily equivalet to the there is a uitary operator hus, U L(H ) such that S U U which iplies that S U U S S U UU US U UU UU U ad ( ) S S ( U UU U ) ( U U ) ( U U ) ( U U ) U ( ) U Sice * ( ) we have * ( ) S S S S U ( ) U hus S power class( (iii) By [] we have ( ) M M M M M ( ) M M M hus powerclass( M he followig eaple shows that if uitarily equivalece i theore (ii) is replaced by siilarity the the result is eed ot be true 0 Eaple Cosider the two operators ad X actig o power class Q Now the two diesioal Hilbert space the ( )

3 O power class ( operators 55 X ad by direct decopositio we show that XX S (say) Now agai by direct decopositio we show that S S while ( S S ) hus S is siilar to but S power class( he followig eaple shows that the su ad product of power class( operators are ot power class( i i 0 Eaple 3 Cosider the operators S ad are power class( operators o the cople Hilbert space But S + ad S are power class Q ot ( ) If powerclass( Reark 4 such that 0 the it is ot ecessarily that 0 Cosider 0 0 actig o 0 heore 5 If L(H) R which is ot oral is -oral the powerclass( Proof Sice is -oral the Pre ultiply by ad post ultiply by o both sides we get, powerclass ( ) Hece ( he followig eaple shows that a operator of power class( eed ot be oral Eaple 6 If 0 0 be a operator actig o three diesioal cople 0 Hilbert space he is powerclass( but it is ot oral

4 56 S Paayappa ad N Sivaai he followig eaples show that a powerclass( eed ot be 3 powerclass( ad vice versa i Eaple 7 Cosider the operator actig o diesioal cople Hilbert space which is powerclass( but ot 3 power class( Eaple 8 Cosider the operator actig o diesioal Hilbert 0 space which is 3powerclass( but ot power class( heore 9 If is power class( ad is quasi oral the is + power class( Proof If is power class( Post ultiply by the * ( ) o both sides ( ) ( )( ) * + Sice is quasi oral we have * Hece ( ( ) ( ) ( ) * ( + ) + + power class he followig eaple shows that the coditio that is quasi oral is ecessary i Eaple 0 Cosider the operator actig o R which is power class(, ot quasi oral ad ot 3 power class( heore Let,, be oral operators i he power class Q operators ad are ( ) Proof ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( )

5 O power class ( operators 57 ( ) ( ) ( )( ) ( ) Hece is ( ) Q power class Now,,, ( ) ( ) ( )( )( ) ( )( )( ) ( ) Hece is ( ) Q power class Refereces [] SAAlzuraiqi, ABPatel, O Noral Operators, Geeral Matheatics Notes, Vol, No, Dec 00, pp6-73

6 58 S Paayappa ad N Sivaai [] AAS Jibril, O Operators for which ( ) * Matheatical Foru, 5,00, 46, 55 6, Iteratioal [3] AAS Jibril, O power oral Operators, he Arabia Joural for Sciece ad Egieerig Volue 33, Nuber A [4] AAS Jibril, O oral Operators Dirasat, Vol3, No (996), [5] Ould Ahed Mahoud Sid Ahed, O he Class of -Power Quasi-oral Operators O Hilbert Space, Buliti of Matheatical Aalysis of Applicatios, Volue 3 Issue (0), PP3-8 Received: Jauary, 0