ANOTHER VERSION OF FUGLEDE-PUTNAM THEOREM

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1 ANOTHER VERSION OF FUGLEDE-PUTNAM THEOREM SALAH MECHERI AND AISSA BAKIR Abstract. In [7] the author proved that, for a M-hyponormal operator A for a dominant operator B, CA = BC implies CA = B C. In the case where A B are normal, this result are known as the Fuglede-Putnam theorem. In this paper, we will extend this result in the case in which A is an injective (p, k)- quasihyponormal operator B is a dominant operator. We also show that the same result remains hold for (p, k)-quasihyponormal log-hyponormal operators.. Introduction For complex infinite dimensional Hilbert space H, B(H) denote the algebra of all bounded linear operators on H. The familiar Fuglede- Putnam theorem is as follows (see [3], [5] [6]): Theorem.. If A B are normal operators if X is an operator such that AX = XB, then A X = XB. Many Authors have extented this theorem for several classes of operators for example (see [], [5], [0]). In [7] the author proved that, for M-hyponormal operator A for dominant operator B, CA = BC implies CA = B C. Recently I.H.Kim [9] showed that Fuglede- Putnam s theorem remains hold for injective (p, k)-quasihyponormal operator log-hyponormal operator. More generally, in this paper it is shown that if A is an injective (p, k)-quasihyponormal in H B is a dominant operator in H such that AC = CB for some C B(H), then A C = CB. We also show that the same result remains hold for injective (p, k)-quasihyponormal log-hyponormal operators. A is said to be (p, k)-quasihyponormal if A k ((A A) p (AA ) p )A k 0 (0 < p, k N), if p =, k= p = k =, then A is k-quasihyponormal, p-quasihyponormal quasihyponormal respectively. A is normaloid if A = r(a) (the spectral radius of A). Let (N), (HN), Q(p), (Q(p, k)) (NL) denote the classes of normal, hyponormal, p-quasihyponormal, (p, k)-quasihyponormal, normaloid This research was supported by KSU research center Project No Mathematics Subject Classification. 47A0,47B0. Key words phrases. Fuglede-Putnam theorem; (p,k)-quasihyponormal operator, Dominant operator.

2 S. MECHERI AND AISSA BAKIR operators. These classes are related by proper inclusions: (N) (HN) (Q(p)) (Q(p, k)) (NL). (see [] ) An operator A B(H) is called dominant by J.G.Stampfli B.L.Wadhwa [] if, for all complex λ, ran(a λ) ran(a λ), or equivalently, if there is a real number M λ such that (A λ) f M λ (A λ)f, for all f H. If there exists a real number M such that M λ M for all λ, the dominant operator A is said to be M- hyponormal. A -hyponormal is hyponormal. Let (D) (m H) denote the classes of dominant M-hyponormal operators. Then (N) (H) (m H) (D). A is said to be p-hyponormal if (A A) p (AA ) p 0 for some 0 < p. If p =, A is said to be hyponormal if p =, A is said to be semihyponormal. A is said to be log-hyponormal if A is invertible satisfies the following equality log(a A) log(aa ). It is known that invertible p-hyponormal operators are log-hyponormal operators but the converse is not true [4]. However it is very interesting that we may regards log-hyponormal operator are 0-hyponormal operator [7]. The idea of log-hyponormal operator is due to Ando [] the first paper in which log-hyponormality appeared is [4]. For properties of log-hyponormal operators (see [, 7, 5]).. Main results Theorem.. Let A B(H) be an injective (p, k)-quasihyponormal operator, let B be log-hyponormal in B(H). If AC = CB for some C B(H), then A C = CB. Proof. Since ranc is invariant under A (ker C) is invariant under B, we can consider the following decompositions H = ranc ranc = (ker C) (ker C) we have A A A = B 0, B = 0 A 3 B B 3 C 0 C = : (ker C) (ker C) ranc ranc From AC = CB, we obtain A C = C B (.)

3 ANOTHER VERSION OF FUGLEDE-PUTNAM THEOREM 3 Let B = U B be the polar decomposition of B. Let B = B U B be the Aluthge transform of B B = B V B be the second Aluthge transform of B where B = V B. From (.) by using the fact that V B = B V, we get A C = C B V (.) Multiply the two members of (.) by B, we obtain ( ) ) ) A C B = (C B B V B = (C B According to [5], B is hyponormal. Thus B is p-hyponormal the operator A is injective (p, k)-quasihyponormal [9]. Then the pair (A, B ) satisfies the Fuglede-Putnam theorem by [9, Theorem ]. Therefore A ran(c B B ) ker(c B are normal operators. ) Since C is injective with dense range B is one to one ) ran (C B = ran(c ) = ran(c) B ker ) (C B = ker(c ) = ker(c) It follows that A B are normal, according to [5], B is also normal. Since A is (p, k)-quasihyponormal its restriction A is normal, ran(c) reduces A [9]. Thus A = 0. Similarily, B is loghyponormal B = B ker(c) is normal, therefore ker(c) reduces B [5]. Hence B = 0. Since the pair (A, B ) satisfies the Fuglede- Putnam s theorem, A C = C B. Consequently A C = CB. Corollary.. A B(H) is normal if only if A is injective (p, k)- quasihyponormal A is log-hyponormal. Theorem.. Let A B(H) be such that A is p-hyponormal let B B(H) be a dominant operator such that CA = CB for some C B(H). Then CA = CB. Proof. Let A B(H) be such that A is p-hyponormal let B B(H) be a dominant operator. For this we consider two cases: Case ( < p ) Since ranc is invariant under B (KerC) is invariant under A, the operators A, B C can be written on the following decompositions of H H = (kerc) (KerC) = ran(c) ranc

4 4 S. MECHERI AND AISSA BAKIR as follows: A = A 0, B = A A 3 Thus from CA = BC we obtain, [ B B C 0, C = 0 B 4 C A = B C. (.3) Let A = U A be the polar decomposition of A. Since U A = A U, the equality (.3) becomes Multiply the two members of (.4) by A C A U = B C. (.4) ] in right we get (C A ) A U( A ) = B (C A ). Thus the Aluthge transform à = A U A is hyponormal for ( < p ), by [] B is dominant []. hence the pair (Ã, B ) satisfies Fuglede-Putnam theorem by [7]. Therefore the restrictions B ran(c A ) à [Ker(C A )] are normal operators. Since C is a one to one mapping with dense range A is a one to one mapping, it follows that : (kerc) (kerc) ranc ranc. ran(c A ) = ran(c ) = ran(c) Ker(C A ) = Ker(C ) = KerC. Hence à is normal by [6]. Therefore A is normal by [3]. Since B is dominant by [] the restriction B is normal, ran(c) reduces B [], similarly, since A is p- hyponormal the restriction B is normal, [KerC] reduces A [9]. Since the pair (A, B ) satisfies Fuglede-Putnam theorem, C A = BC. This implies that CA = B C. Now concerning the case (0 < p ), it suffices to take p = p +, where p (, ]. It comes back that A is p -hyponormal the proof can be achieved by the same way as the first part. Theorem.3. Let A B(H) be an injective (p, k)-quasihyponormal operator, let B be dominant in B(H). If AC = CB for some C B(H), then A C = CB. For the proof, we need the following lemma. Lemma.. If T is dominant in B(H), then ker(t λ) k = ker(t λ), (λ C, k N ). Proof. For k =. It is clear that ker(t λ) ker(t λ) k. Let x ker(t λ). Then, (T λ)x ker(t λ). Since T is dominant, (T λ)x ran(t λ) ran(t λ) = ker(t λ). It follows that (T λ)x = 0.

5 ANOTHER VERSION OF FUGLEDE-PUTNAM THEOREM 5 Proof of Theorem.. Since ranc is invariant under A (ker C) is invariant under B, we can write H = ranc ranc = (ker C) (ker C) A A A = B 0, B = 0 A 3 B B 3 C 0 C = : (ker C) (ker C) ranc ranc From AC = CB, we have A C = C B, (.5) where A is injective (p, k)-quasihyponormal by [9], B is dominant [] C : (ker C) ranc is injective with dense range. Let us consider the decompositions Then ranc = ran(a k ) ker(a k ) (ker C) = ran(b k ) ker(b k ) = ran(b k ) ker(b ) A = A A C C, C 0 A = 3 C 3 C 4 B = [ B 0 because ker(b ) reduces (B ). According to [9], A is injective p- hyponormal on ran(a k ), A k 3 = 0. B is dominant. From (.5), we obtain A C +A C 3 = C B ; A C +A C 4 = 0 ; A 3 C 3 = 0 A 3 C 4 = 0 (.6) A C = C B implies A k C = C B k. Thus C ran(b k ) = ran(a k ). Since C : ran(b k ) ran(a k ) C 3 : ran(b k ) ker(a k ) then C 3 = 0. Hence (.6) implies A C = C B. Since the pair (A, B ) has the (F P ) property, A C = C B [ A ranc ], B (ker C ) are normal operators by []. Since C C C = is 0 C 4 injective with dense range, it follows that C is injective with dense range. Therefore ranc = ran(a k ) (ker C ) = {0} = ran(b). k Hence A B are normal. Then we have A A C = C A C A C A C + A ; C B 3C C B = 0. 4 ]

6 6 S. MECHERI AND AISSA BAKIR Since A = A ranc is injective normal opertor A is (p, k)- quasihyponormal, ranc reduces A by [9]. Therefore A[ = 0. Hence, ] C 0 A C = 0 implies C = 0. The operator C = is 0 C 4 injective with dense range, consequently, C is injective. Hence, C4 is a one to one mapping. Therefore, A 3 C 4 = 0 implies [ C4A 3 ] = 0 so A 3 = 0. Hence A C = C B. A 0 Since A = B 0 B = are normal, ranc reduces A (ker C) reduces B. Hence A = B = 0. Finally, A C = CB. Corollary.. A B(H) is normal if only if A is injective (p, k)- quasihyponormal A is dominant. Now if we take in Theorem., A log-hyponormal B (p, k)- quasihyponormal, we can ask does Theorem. remains hold in this case. in the following theorem we will show that the answer is positive. Theorem.4. Let A B(H) be such that A is log-hyponormal let B B(H) be injective (p, k)-quasihyponormal. If AC = CB for some C B(H), then A C = CB. Proof. For log-hyponormal operator A injective (p, k)-quasihyponormal operator B, AC = CB implies A C = CB by the previous theorem. Put A = B, B = A C = C. Then A is injective (p, k)- quasihyponormal B is log-hyponormal. Moreover A C = C B implies A C = C B. Which completes the proof. By the same arguments we prove that Theorem.3 remains hold if we take A dominant B (p, k)-quasihyponormal. Remark.. In [9, Lemma 0], the injectivity condition can be dropped if A is p-hyponormal instead of (p, k)-quasihyponormal (see [8, Lemma ]). Therefore we recapture a generalized Fuglede-Putnam s theorem for p-hyponormal operator. Corollary.3. [7, Theorem 8] Let A be p-hyponormal let B be log-hyponormal. If AC = CB for some C B(H), then A C = CB. References [] A. Aluthge D.Wang, An operator inequality which implies paranormality, Math. Inequalities Applications, (999), 3-9. [] T.Ando, operators with a norm condition, Acta. Sci. Math(Szeged), 33(97), [3] J.B.Conway, Subnormal operators, Research notes in Math., Pitnam Advanced Pub. Program, 5(98). [4] M. Fujii, C. Himeji A. Matsumoto, Theorems of Ando Saito for p- hyponormal operators, Math. Japonica, 39(994), [5] T.Furuta, Invitation to linear operators, Taylor Francis Inc., (00).

7 ANOTHER VERSION OF FUGLEDE-PUTNAM THEOREM 7 [6] P.R.Halmos, A Hilbert space problem book, spring-verlag, New York, (974). [7] I.H.Jeon, K. Tanahashi, A. Uchiyama, On quasisimilarity for loghyponormal operators, Glasgow Math. J, 46(004), [8] I.H. Jeon, p-hyponormal operators quasisimilarity, integr. Equat. Operat. theor, 49(004), [9] I.H.Kim, The Fuglede-Putnam s theorem for (p, k)-quasihyponormal operators, J. Ineq. Appl, (006), article ID 4748, page -7. [0] S. Mecheri, K.Tanahashi, A. Uchiyama, Fuglede-Putnam theorem for p- hyponormal or class Y operator, Bull Korean. Math. Soc, 43(006), [] M.Y.Lee, An extension of the Fuglede-Putnam theorem to (p, k)- quasihyponormal operator, Kyungpook Math.J., 44(004), [] J.G.Stampfli B.L. Wadhwa. An asymmetric Putnam-Fuglede theorem for dominant operators, Indiana Univ. Math. J., 5(976)., [3] H.Tadashi, A note on p-hyponormal operators., Proc.Amer.Math.Soc, 5(997), -30. [4] K.Tanahashi A. Uchiyama, Integr. Equat. operat. Theor, 34(999), [5] A.Uchiyama K.Tanahashi, Fuglede-Putnam s theorem for p-hyponormal or log-hyponormal operators, Glaskow Math.J., 44(00), [6] A.Uchiyama, Berger-Shaw s theorem for p-hyponormal operators, Integral equations Operator theory., Inequalities in operator theory related topics (Japanese) (Kyoto, 997). Sūrikaisekikenkyūsho Kōkyūroku No. 07, (998), 0-. [7] T. Yoshino, Remark on the Generalized Putnam-Fuglede theorem, Proc.Amer.Math.Soc., 95(985), Salah Mecheri Department of Mathematics King Saud University, College of Science P.O.Box 455, Riyadh 45, Saudi Arabia. Aissa Bakir Department of Mathematics Mostaganem University, College of Science Mostaganem, Algeria. address: mecherisalah@hotmail.com address: aissabakir@yahoo.fr