A FUGLEDE-PUTNAM TYPE THEOREM FOR ALMOST NORMAL OPERATORS WITH FINITE k 1 - FUNCTION

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1 Research ad Commuicatios i Mathematics ad Mathematical Scieces Vol. 9, Issue, 07, Pages 3-36 ISSN Published Olie o October, Jyoti Academic Press A FUGLEDE-PUTNAM TYPE THEOREM FOR ALMOST NORMAL OPERATORS WITH FINITE k - FUNCTION Departmet of Mathematics Florida A&M Uiversity Tallahassee FL 3307 USA vasile.lauric@famu.edu Abstract I this ote, we will prove that operators T L( H) with fiite k fuctio satisfy a Fuglede-Putam type modulo the Hilbert-Schmidt class, that is, for arbitrary X L( H) with TX XT C ( H) implies T X XT C ( H ).. Let H be a separable, ifiite dimesioal, complex Hilbert space, ad deote by L ( H) the algebra of all bouded liear operators o H ad by C p ( H) (or simply C p ) the Shatte-vo Neuma p-classes ad by p, p, their respective orm. I this ote oly the particular classes correspodig to p =, will be used, that is the trace-class C ad the 00 Mathematics Subject Classificatio: Primary 47B0, 47B37. Keywords ad phrases: almost ormal operators, k - fuctio, Fuglede-Putam type theorem. Received August 3, 07

2 3 class of Hilbert-Schmidt operators C. For arbitrary operators S, T L( H), [ S, T ] will deote their commutator ST TS ad D S will deote the self-commutator of S, that is [ S, S ]. A operator S L( H) is called almost ormal whe D S C ( H) ad the class of operators defied o H which are almost ormal will be deoted by AN ( H).. Voiculescu s Cojecture 4 ( C 4 ), (cf. [3] or [4]) states that for T AN( H), there exists S AN( H) such that T S = N + K, where N is a ormal operator ad K is a Hilbert-Schmidt operator. Uder the assumptio that cojecture ( C 4 ) has a positive aswer, oe ca easily prove that almost ormal operators satisfy a Fuglede-Putam type theorem, that is, if T AN ( H) ad X L( H) is such that [ T, X ] C ( H), the [, X ] C ( H), T ad the family of operators o H that have such a property will be deoted by FP ( H). This results from a theorem of Weiss [5] that states that if N L( H) is a ormal operator ad X L( H) such that [ N, X ] C ( H), the [ N, X ] C ( H ) ad [ N, X ] = [ N, X ] (i particular N FP ( H) left for the reader. ), ad the details are + Let P ad R deote the set of fiite rak orthogoal projectios ad the fiite rak positive semidefiite cotractios, respectively, ad qp ( T ) = lim if (( I P ) TP ), k p P P ( T ) = lim if [ T, A], + A R where the lim ifs are with respect to the atural order. I [], it was proved that almost ormal operators T such that q ( T ) < belog to FP ( ). H p p

3 A FUGLEDE-PUTNAM TYPE THEOREM FOR It is atural to ask whether almost ormal operators T with k ( T ) fiite, ad implicitly k ( T ) 0 (cf. []), belog to F. = We will prove that such a result holds uder the hypothesis that k ( T ) is fiite. Theorem. If T AN ( H) ad k ( T ) <, the T FP ( H). Proof. Let T AN ( H) with k ( T ) <, let A + R,, so that P A I ad [ T ] k ( ), ad let X L( H) with [ T, X ] = : R C ( H). It will be eough to prove that lim sup where Q : = T X XT. A, T tr[ ( QQ RR )] <, A Write A RR = ATXX T ATXT X AXTX T + AXTT X = a b c + d ad A QQ = AT XX T AT XTX AXT X T + A XT TX = A B C + D, where a, b,, C, D are the terms i the order they appear i these expasios. First The tr ( D d) D d X DT. () tr ( B c) = tr( A T XTX A XTX T ) ( A T XTX T A XTX ) = tr([ A T ] ) = tr, XTX [ A, T ] XTX [ A, T ] XTX [ A, T ] X, T

4 34 ad the after passig to limit I a similar way, T tr ( C b) = tr( A XT X T A TXT X ) tr( B c) k ( T ) X. () ( TA XT X T A TXT X ) = tr([ T A ] XT ) = tr, X [ A, T ] XT X [ A, T ] XT X [ A, T ] X, T ad thus T tr( C b) k ( T ) X. (3) Fially, tr ( A a) = tr( A T XX T A TXX T ) = tr( TA T XX T A TXX ) TA T T A T X. Furthermore, TAT T AT = TAT ATT AT T + AT T T AT = [[ T, A ] T + ADT + [ A, T ] T [ T, A ] T + D + [ A T ] T T, [ T, A ] T + D, T ad cosequetly, by passig to limit, we have Usig iequalities ()-(4), T X tr( A a) ( k ( T ) T + D ). (4)

5 A FUGLEDE-PUTNAM TYPE THEOREM FOR 35 lim sup tr[ A ( QQ RR )] 4k ( T ) X T + D, which eds the proof. The above proof leads to the followig. T X Corollary. If T AN ( H) with k ( T ) < ad X L( H) so that [ T, X ] C ( H), the [ T X ] [ T, X ] + 4k ( T ) X T + D. T X Corollary 3. If T, S AN ( H) with k ( T ) ad k ( S) < ad X L( H) so that R : = TX XS C ( H), the Q : = T X XS C ( H). that Proof. Let T, S, X as i the hypothesis. It is straightforward to see k ( T S) k ( T ) + k ( S) <. 0 ~ X 0 Settig ~ ~ R X =, the ( T S) X X ( T S) =, ad ~ ~ ~ ~ thus ( T S) X X ( T S) C. Therefore ( T S) X X ( T S) = 0 0 Q C. 0 Cosequetly, Q C ( H). Corollary 4. If T AN ( H) with k ( T ) < ad X L( H) so that TX XT C ( H), the T X XT C ( H). Refereces [] Vasile Lauric, A Fuglede-Putam theorem modulo the Hilbert-Schmidt class for almost ormal operators with fiite modulus of Hilbert-Schmidt quasi-triagularity, Cocr. Oper. 3 (06), 8-4. [] Da V. Voiculescu, Some results o orm-ideal perturbatios of Hilbert space operators II, J. Operator Theory 5 (98),

6 36 [3] Da V. Voiculescu, Hilbert space operators modulo ormed ideals, Proceedigs of the Iteratioal Cogress of Mathematicias, August 6-4, Warszawa (983), [4] Da V. Voiculescu, Almost ormal operators mod Hilbert-Schmidt ad the K-theory of the algebras E Λ( Ω), J. Nocommut. Geom. 8 (04), [5] Gary Weiss, The Fuglede commutativity theorem modulo operator ideals, Proc. Amer. Math. Soc. 83 (98), 3-8. g