ON THE GENERALIZED FUGLEDE-PUTNAM THEOREM M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI

TAMKANG JOURNAL OF MATHEMATICS Volume 39, Number 3, 239-246, Autumn 2008 0pt0pt ON THE GENERALIZED FUGLEDE-PUTNAM THEOREM M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI Abstract. In this paper, we prove the following assertions: If the pair of operators A,B satisfies the Fuglede-Putnam Property and S kerδ A,B, where S BH, then we have δ A,B X + S S. 2 Suppose the pair of operators A,B satisfies the Fuglede-Putnam Property. If A 2 X X B 2 and A 3 X X B 3, then AX X B. 3 Let A,B BH be such that A,B are p-hyponormal. Then for any X C 2, AX X B C 2 implies A X X B C 2. 4 Let T,S BH be such that T and S are quasihyponormal operators. If X BH and T X X S, then T X X S.. Introduction Let H denote a separable, infinite dimensional Hilbert space. Let BH, C 2 and C denote the algebra of all bounded operators acting on H, the Hilbert-Schmidt class and the trace class in BH respectively. It is known that C 2 and C each form a two-sided ideal in BH and C 2 is itself a Hilbert space with inner product X,Y X e i,y e i trx Y, where {e i } is any orthonormal basis of H and tr. is the natural trace on C. The Hilbert- Schmidt norm of X C 2 is given by X 2. For operators A,B BH, the generalized derivation δ A,B X as an operator on BH is defined as follows: δ A,B X AX X B for all X BH. When A B, we simply write δ A for δ A,A. In [3], Anderson proved that if N BH is normal, S is an operator such that N S SN, then for all X BH δ N X + S S, Received April 30, 2007. 2000 Mathematics Subject Classification. 47A0, 47B20, 47B47, 47A30. Key words and phrases. Normal derivation, quasihyponormal, Hilbert Schmidt operator, p-hyponormal, Fuglede-Putnam Theorem. 239

240 M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI where. is the usual operator norm. Thus in the sense of [3, Definition.2], Anderson s result says that the range of δ N is orthogonal to the kernel of δ N, which is just the commutant {N } of N. Kittaneh [0] extended this result to an arbitrary unitarily invariant norm for more information about this norm the reader should refer to [0], and he proved the following theorem. Theorem.. If the operator A,B BH are normal, then for all X,S BH such that AS SB we have AX X B + S S. Recall that an operator T in BH is called normal if T T T T and p-hyponormal if T T p T T p, where 0 < p. In particular -hyponormal is called hyponormal and 2 -hyponormal is called semi-hyponormal. The Löwner-Heinz inequality implies that if T is p hyponor mal, then it is q hyponor mal for any 0 < q p. Let T U T be the polar decomposition of T, where U is partial isometry. Then T is a positive square root of T T and kert ker T keru, where kert denotes the kernel of T. Aluthge [2] introduced the operator T T 2 U T 2 which is called the Aluthge transform, and also shown to satisfy the following result. Theorem.2. Let A U A be the polar decomposition of a p-hyponormal for 0 < p < and U is unitary. Then the following assertion holds: Ã A 2 U A 2 is p + 2 -hyponormal if 0 < p < 2. 2 Ã A 2 U A 2 is hyponormal if 2 p <. According to [3], T BH is called dominant if r ant zi r ant zi, for all z σt, where r ant and σt denotes the range and the spectrum of T. It has been shown in [4, Theorem 3.4] that if A, B and S are operators in BH such that B is p-hyponormal or log-hyponormal, A is dominant and AS SB, then for all X BH we have AX X B + S S. 2. Main Results We begin by the following definition of the orthogonality in the sense of Anderson [3, Definition.2] which generalize the idea of orthogonality in Hilbert space. Definition 2.. Let C be the field of complex numbers and let X be a normed linear space. If x λy λy for all λ C whenever x, y X, then x is said to be orthogonal to y. Let U and W be two subspace in X. If x + y y, for all x U and for all y W, then U is said to be orthogonal to W.

ON THE GENERALIZED FUGLEDE-PUTNAM THEOREM 24 Definition 2.2. Let A,B BH. We say that the pair A,B satisfies the Fuglede-Putnam Property, if whenever S kerδ A,B, where S BH implies that S kerδ A,B. Theorem 2.3. Let A,B, X BH. If the pair of operators A,B satisfies the Fuglede- Putnam Property and S kerδ A,B, where S BH then we have δ A,B X + S S. Proof. Since the pair A,B satisfies the Fuglede-Putnam Property, it follows that r ans reduces A, ker S reduces B. Let A A r ans, B B ker S and let S : ker S r ans be the quasi-affinity defined by setting S x Sx for each x ker S. Then δ A,B S 0 δ A,B S, and it follows that 0 σa and 0 σb. Since the pair A,B satisfies the Fuglede-Putnam Property, then δ A,B AS 0 implies A B S A S B A A S, then A is normal, similarly B is normal. Then, with respect to the orthogonal decompositions H r ans r ans and H ker S A 0 B 0 kers, A and B can be respectively represented as A, B. 0 A 2 0 B 2 Now assume that the operators S, X : ker S kers r ans r ans have the matrix representations and. S 0 X X 2 X 3 X 4 Then δa,b δ A,B X + S X + S δ A,B X + S S, which completes the proof of the theorem. Next, we prove some commutativity results. Al-Moadjil [] proved that if N is a normal operator such that N 2 X X N 2 and N 3 X X N 3 for some X BH, then N X X N. Kittaneh [7] generalize this results for subnormal operators by taking A and B subnormal operators. This result was also generalized by Bachir [4] by taking A is a dominant operator and B is P-hyponormal operators. In this note, we generalize this result for any pair A,B satisfying the Fuglede-Putnam Property. Theorem 2.4. Let A,B, X BH and let the pair of operators A,B satisfies the Fuglede- Putnam Property. If A 2 X X B 2 and A 3 X X B 3, then AX X B. and Proof. Let Y AX X B, then A 2 Y A 3 X A 2 X B X B 3 X B 3 0, Y B 2 AX B 2 X B 3 A 3 X A 3 X 0, AY B A 2 X B AX B 2 X B 3 A 3 X 0. Hence AAY Y B A 2 Y AY B 0 and AY Y BB AY B Y B 2 0. This yields that AY Y B kerδ A,B r anδ A,B {0}, therefore AY Y B 0. Hence Y

242 M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI kerδ A,B r anδ A,B {0} is obtained by Theorem 2.3, this implies that Y 0. That is, AX X B. Kittaneh [7] proved that if A,B BH such that A 2 B 2, A 3 B 3, ker A ker A and kerb kerb, then A B. This results can be generalized to any pair A,B that satisfies the Fuglede-Putnam Property as follows. Corollary 2.5. Let A,B BH. If the pair of operators A,B satisfies the Fuglede-Putnam Property and A 2 B 2 and A 3 B 3, then A B. Proof. This is an immediate consequence of Theorem 2.3 and Theorem 2.4. Remark 2.6. Algebraic manipulations and induction show that the powers 2 and 3 in Theorem 2.4 and Corollary 2.5 can be replaced by any two relatively prime powers n and m. In order to obtain our next result, we require some lemmas. T S Lemma 2.7. Let T be a quasihyponormal operator on H M M, where M is 0 T 2 a T -invariant such that the restriction T T M is normal. Then the range of S is included in kert. In particular, if T is injective, every normal part of T reduces T. Proof. Let P be the orthogonal projection onto M. Then we have Let T 2 T T. Since Hence PT T P T T 0 P T 2 P T 2T 2 2 0 T T 0 since T is quasihyponormal by Hansen s inequality since T is normal. X Y Y be the matrix representation of T 2 on H M M. Then we have X Z T 2 2 T 2 T 2 This implies that Y 0. Thus we have T 2 T T 0 0 Z 2 X 2 + Y Y X Y + Y Z T 2 Z Y + Y X Y Y + Z 2 X 2 + Y Y T 2 T 2 T T 2 X 2. T 2 T 2T S S T T 2 S S + T2 2T 2 2 T T T T T S S T S S + T2 T, 2.

ON THE GENERALIZED FUGLEDE-PUTNAM THEOREM 243 and hence T S 0. Thus the range of S is included in kert kert. If T is injective, then so is T. Thus the second statement of lemma follows from [6, Lemma 0]. Lemma 2.8. If T is a quasihyponormal operator, then every normal part of T reduces T. Proof. If T is invertible, then T is hyponormal. Hence the assertion holds by Stampfli result [2]. Now, we assume that T is not invertible. Let M be a normal part of T. By Lemma 2.7 N S T is of the form on M M, where N is normal and r ans ker N. It is easy to see that and Then This implies that S 0. T 2 T 2 N 2 N 2 N 2 N S S N N 2 S N N S T T 2 N N 2 + N SS N N N N S + N SS S S N N N + S SS N S N N S + S SS. S 0 T 2 T 2 T T 2 N SS N N SS S S SS N S SS. S Theorem 2.9. Let T BH be a quasihyponormal operator. Let L BH be self-adjoint which satisfies T L LT. Then T L LT. Proof. First, we will show that If T L LT 0, then T L LT 0. Since kert reduces T, T L 0 implies that r anl kert kert. Hence r ant kert. Therefore we have T L LT 0 Next, we prove the case T L 0. Assume that T is quasihyponormal. Using the decomposition H r anl kerl, the operators L and T can be represented as follows. L L 0 T S,T, 0 T 2 where L is self-adjoint with kerl {0} and T is also quasihyponormal. The assumption T L LT implies that T L L T. Since kert reduces T and L, they are of the form T T 0 and L L L 22 on r anl ker T kert. Hence T is an injective quasihyponormal operator and L is self-adjoint operator which satisfies T L L T. But this implies that T is normal. Hence T T 0 is also normal. By Fuglede-Putnam Theorem, we see that T L L T. since T is normal, S 0, so we have T L LT. Corollary 2.0. Let T BH be a quasihyponormal operator. If X BH and T X X T, then T X X T. Proof. Let X Y + i Z be the cartesian decomposition of X. Then T X X T implies that T Y Y T and T Z Z T. By Theorem 2.9, we have T Y Y T and T Z Z T. Hence T X X T.

244 M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI Corollary 2.. Let T,S BH be such that T and S are quasihyponormal operators. If X BH and T X X S, then T X X S. T 0 0 X Proof. Let A 0 S and B on H H. Then A is quasihyponormal operator on H H that satisfies 0 T X 0 X S AB B A, by Corollary 2.0, we have A B B A and therefore T X X S. The most recent generalization of the Fuglede-Putnam theorem was obtained by Uchiyama and Tanahashi [5] and can be stated as follows. Theorem 2.2. Let T,S BH be such that T and S are p-hyponormal operators. If X BH and T X X S, then T X X S. As an application of the above results we have Lemma 2.3. Let V, A and X be operators in BH. If V is an isometry, A is p-hyponormal, and X is one-one, then V X X A implies A is unitary. Proof. By Corollary, V X X A implies that V X X A. Multiply the first equation on the left by V to get X V A A. Therefore, X X A A. Let X U P be the polar decomposition of X, then U is unitary and P is one-one. But this implies that A A. Since A is p-hyponormal and A A, it follows that A is normal and hence unitary. An attempt to generalize Theorem [7, Theorem ] to the hyponormal case was made by T. Furuta [5], who obtained the following result. Theorem 2.4. If A and B are hyponormal operators in BH, then for any X C 2, AX X B C 2 implies A X X B C 2. In the following theorem, we relax the hypotheses on A and B in Theorem 2.4 to p- hyponormality. Theorem 2.5. Let A,B BH be such that A,B are p-hyponormal. Then for any X C 2, AX X B C 2 implies A X X B C 2. Proof. Let A,B be p-hyponormal for p 2 and let U B be the the polar decomposition of B. Then it follows from [2] that the Aluthge transform B of B is hyponormal and satisfies B B B. and AY Y B C 2.2

ON THE GENERALIZED FUGLEDE-PUTNAM THEOREM 245, where Y XU B 2. Using the decomposition H kery kery, we see that A, B,Y are of the form A T B 0 Y 0 A, B,Y, 0 A 2 S B 2 where, A is p-hyponormal, B is hyponormal and Y is one-one mapping with dense range. It follows from equation.2 that A Y Y B C 2..3 Hence A,B are normal by [, Theorem 0], so that T 0 by [5, Lemma 2] and S 0 by [6]. Thus B B J, for some positive operator J, by equation. and [5, Lemma3] that U U 2 X X 2 U. Let X be a 2 2 matrix representation of X with respect to the 0 U 22 X 2 X 22 decomposition H kery kery. Then, Y XU B 2 implies that Y X U B 2 and hence kerb kery {0}. This shows that B is one-one hence, it has dense range, so that U 2 0 and B B B 3, for some co-p-hyponormal operator B 3 by [5, Lemma 3]. Since, we have the following statements. Y 0 Y XU B X X 2 2 U B 2 0 X 2 X 22 0 U 22 A 33, 2 X 2 U 22 B 3 2 0; hence X 2 B 3 0 because B 3 U 22 B 3. X 2 U B 2 0; hence X 2 0 because U B 2 has dense range. X 22 U 22 B 3 2 0; hence X 22 B 3 0. The assumption AX X B C 2 imply that A X X B C 2, X 2 B 3 A X 2 0 and X 22 B 3 A 2 X 22 0. Since A and B are normal we have A X X B C 2, by Fuglede-Putnam Theorem. The p-hyponormality of B3 shows that r anb3 B 3. Also we have ker A 2 ker A 2 from the fact A 2 is p-hyponormal. Hence, we also have X 2 B3 A X 2 0 and X 22 B3 A 2 X 22 0. This implies that AX X B C 2. Next, we prove the case where 0 < p 2. Let Y be as above. Then B is p + 2 -hyponormal and satisfies AX X B C 2. Use the same argument as above. We obtain B B B 2 on H kery kery and A A A 2, where B is an injective normal operator and A is also normal. Hence, we have B B B 3 for some p-hyponormal B3. Again using the same argument as above we obtain the result. References [] A. Al-Moadjil, On the commutant of relatively prime powers in banach algebra, Proc. Amer. Math. Soc. 57976, 243 249. [2] A. Aluthge, On the p-hyponormal for 0 < p < 2, Integ. Equat. Oper. Th. 3990, 307 35. [3] J. Anderson, On normal derivations, Proc. Amer. Math. Soc. 38973, 35 40. [4] A. Bachir, Generalized derivation, SUT J. Math 402004, no.2, -4. [5] T. Furuta, An extension of the Fuglede-Putnam Theorem to subnormal operators using a Hilbert- Schmidt norm inequality, Proc. Amer. Math. Soc.898, 240 242.

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