THE FUGLEDE-PUTNAM THEOREM FOR

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1 THE FUGLEDE-PUTNAM THEOREM FOR (p,k-quasihyponormal OPERATORS IN HYOUN KIM Received 8 September 24; Accepted 19 September 24 We show that if T ( isa(p,k-quasihyponormal operator and S ( isaphyponormal operator, and if TX XS,whereX : is a quasiaffinity (i.e., a one-one map having dense range, then T is a normal and moreover T is unitarily equivalent to S. Copyright 26 Hindawi Publishing Corporation. All rights reserved. Let be a separable complex Hilbert space with inner product, and let ( denote the C -algebra of all bounded linear operators on. The spectrum of an operator T, denoted by σ(t, is the set of all complex numbers λ for which T λi is not invertible. ThenumericalrangeofanoperatorT, denoted by W(T, is the set defined by W(T { Tx,x : x 1 }. (1 The norm closure of a subspace of is denoted by. We denote the kernel and the rangeofanoperatort by ker(t and ran(t, respectively. For p such as <p 1 and positive integer k, anoperatort ( iscalled(p,k- quasihyponormal if T k ( T 2p T 2p T k. A (p,k-quasihyponormal operator is an extension of p-hyponormal operator (i.e., (T T p ( p, k-quasihyponormal operator (i.e., T k ( T 2 T 2 T k and p-quasihyponormal operator (i.e., T ( T 2p T 2p T. Aluthge [1], Campbell and Gupta [3], Arora and Arora [5], and the author [8] introduced p-hyponormal, k-quasihyponormal, p-quasihyponormal, and (p,k- quasihyponormal operators, respectively. It was known that these operators share many interesting properties with hyponormal operators (see [1 8, 11, 12]. In this paper, we consider the extension of results of Sheth [9] and Gupta and Ramanujan [6]. The main result is as follows. If T ( isa(p,k-quasihyponormal operator and S ( isap-hyponormal operator, and if TX XS,whereX : is an injective bounded linear operator with dense range, then T is a normal operator unitarily equivalent to S. In general, the conditions S 1 TS T and / W(S do not imply that T is normal. For example, (see [13], if T SB,whereS is positive and invertible, B is self-adjoint, and Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 26, Article ID 47481, Pages 1 7 DOI /JIA/26/47481

2 2 The Fuglede-Putnam theorem S and B do not commute, then S 1 TS T and / W(S, but T is not normal. Therefore the following question arises naturally. Question 1. Which operator T satisfying the condition S 1 TS T and / W(S is normal? In 1966, Sheth [9] showed that if T is a hyponormal operator and S 1 TS T for any operator S, where / W(S, then T is self-adjoint. We extend the result of Sheth to the class of p-hyponormal operators as follows. Theorem 2. If T or T is p-hyponormal operator and S is an operator for which / W(S and ST T S, then T is self-adjoint. To prove Theorem 2 we need the following lemma. Lemma 3[13, Theorem1]. If T ( is any operator such that S 1 TS T,where / W(S, then σ(t R. Proof of Theorem 2. Suppose that T or T is p-hyponormal operator. Since σ(s W(S, S is invertible and hence ST T S becomes S 1 T S T (T.ApplyLemma 3 to T to get σ(t R. Thenσ(T σ(t σ(t R. Thusm 2 (σ(t m 2 (σ(t for the planer Lebesgue measure m 2. Now apply Putnam s inequality for p-hyponormal operators to T or to T (depending upon which is p-hyponormal to get or ( T T p ( p p π ( p ( T T p p π σ(t σ(t r 2p 1 dr dθ (2 r 2p 1 dr dθ. (3 It follows that T or T is normal. Since σ(t σ(t R here, T must be selfadjoint. We can extend the result of Theorem 2 to the class of p-quasihyponormal operators. We use the following lemma. Lemma 4[8, Lemma 1]. If T is (p,k-quasihyponormal operator, then T has the following matrix representation: ( T, (4 T 3 where T 1 is p-hyponormal on ran(t k and T 3 k.furthermore,σ(t σ(t 1 {}. Theorem 5. If T is (p,k-quasihyponormal operator and S is an arbitrary operator for which / W(S and ST T S, then T is direct sum of a self-adjoint and nilpotent operator.

3 In Hyoun Kim 3 Proof. Since T is (p,k-quasihyponormal operator, we have the following matrix representation: ( T on ran ( T T k ker ( T k, (5 3 where T 1 is p-hyponormal and T k 3. Since S 1 TS T and / W(S, we have σ(t R by Lemma 3. Therefore σ(t 1 R because σ(t σ(t 1 {} and hence T 1 is selfadjoint by Theorem 2 because T 1 is p-hyponormal operator. Now let P is the orthogonal projection of onto ran(t k. Since T is (p,k-quasihyponormal operator we have p ( TPT p ( P p ( P P T T p ( P PT TP p T p 1 T 1, by Löwner-Heinz s inequality and Hansen s inequality. By Löwner s inequality, for < q p 1, we have q (6 P ( q ( P P T T q T q P 1 T 1. (7 Since T 1 is normal, ( q has the following matrix representation: ( q T1 T q 1 A on ran ( T B k ker ( T k. (8 A Put q p/2. Then by straightforward calculation we have p P ( p ( P P q( q T1 T P 1 which implies A. Thus we have T1 T q 1/q 1 B p + AA, (9 ( B 1/q, (1 and by matrix representation of T we also have ( + T 2 T2 T 2 T3 T 3 T2 T 3 T3. (11 Therefore T 1 T 1 + T 2 T 2 T 1 T 1 and hence T 2, which implies the proof. The following corollary is an extension of the result of Theorem 2 to the class of p- quasihyponormal operators.

4 4 The Fuglede-Putnam theorem Corollary 6. If T or T is p-quasihyponormal operator and S is an arbitrary operator for which / W(S and ST T S, then T is self-adjoint. Proof. If T is p-quasihyponormal operator, T has the following matrix representation by Lemma 4: ( T, (12 where T 1 is p-hyponormal on ran(t k andσ(t σ(t 1 {}. SinceT 1 is self-adjoint and T 2 bytheorem 5, T ( T 1 is also self-adjoint. On the other hand, if T is (p,k-quasihyponormal operator, then using the arguments of the proof of Theorem 2 wecanconcludethatt is self-adjoint. In 1977, Stampli and Wadhwa [1] showed that if A ( ishyponormal,b ( is dominant, C (, is injective and has dense range, and if CA BC, then A and B are normal. On the other hand, in 1981, Gupta and Ramanujan [6] showed that if T ( isk-quasihyponormal operator and S ( is a normal operator for which TX XS where X (, is one to one operator with dense range, then T is normal operator unitarily equivalent to S. In the following theorem, we extend the result ofguptaandramanujantotheclassof(p,k-quasihyponormal operators. We need the followinglemmaduetojeonandduggal[7]. Lemma 7[7,Corollary7]. Let T ( be a p-hyponormal operator and let S ( be a p-hyponormal operator. If TX XS,whereX : is an injective bounded linear operator with dense range then T is a normal operator unitarily equivalent to S. Theorem 8. Let T ( is a (p,k-quasihyponormal operator and let S ( is a p- hyponormal operator. If TX XS,whereX : is an injective bounded linear operator with dense range then T is a normal operator unitarily equivalent to S. Proof. Let T 1 : T ran(tk and S 1 : S ran(sk. Then we have the following matrix representations: ( ( S1 T, S, (13 T 3 where T 1 is p-hyponormal, T k 3 ands 1 is p-hyponormal. Notice that T k X XS k for all positive integer k. ThusX(ran(S k ran(t k. If we denote the restriction of X to ran(s k byx 1 then X 1 : ran(s k ran(t k is one to one and has dense range. Since X 1 S 1 x XSx TXx T 1 X 1 x for every x ran(s k, it follows that X 1 S 1 T 1 X 1.Onthe other hand, since T 1 and S 1 are p-hyponormal operators, it follows from Lemma 7 that T 1 is a normal operator unitarily equivalent to S 1.NowletP be the orthogonal projection of onto ran(t k. Since T is (p,k-quasihyponormal operator and T 1 is normal operator, from the arguments of the proof of the Theorem 5 we have T 2 and hence ran(t k reduces T.SinceX (ker(t k ker(s k ker(s, we have that for each x ker(t k, X T 3 x X T x S X x. (14

5 In Hyoun Kim 5 But since X has dense range, X is one to one and hence T3 x foreveryx ker(t k. Thus T 3, so that T T 1. This completes the proof. Lemma 9[11, Lemma 5]. The restriction T of the (p,k-quasihyponormal operator T on to an invariant subspace of T is also (p,k-quasihyponormal operator. Lemma 1. Let T ( be a (p,k-quasihyponormal operator and be an invariant subspace of T for which T is an injective normal operator. Then reduces T. Proof. Suppose that P is a orthogonal projection of onto ran(t k. Then since T is (p,k-quasihyponormal operator, we have P{(T T p ( p }P. Put T 1 T and T ( T 3 on. (15 Since by assumption T 1 is injective normal operator, we have E P for the orthogonal projection E of onto and ran(t k 1 because T 1 has dense range. Therefore ran(t k and hence E{(T T p ( p }E. Since T is (p,k-quasihyponormal operator, using the Löwner-Heinz inequality and Hansen s inequality we have p E ( TET p ( E E p ( E E T T p ( E ET TE p T p 1 T 1. (16 Since T 1 is normal, we have, by Löwner s inequality, ( p/2 T1 T p/2 1 A. (17 B A So p E ( p T1 T E 1 p + AA, (18 and hence A and ( T 1T 1 B 2/p.Since it follows that T 2 and hence T is reduced by. ( + T 2 T2 T 2 T3 T 3 T2 T 3 T3, (19 Theorem 11. If T ( is p-hyponormal, S ( is injective (p,k-quasihyponormal, and if XT SX for X (,, then XT S X.

6 6 The Fuglede-Putnam theorem Proof. Since by assumption XT SX, we can see that (kerx and ranx are invariant subspaces of T and S, respectively. Therefore by Lemma 9 we have that T (kerx is p-hyponormal and S ranx is also (p,k-quasihyponormal. Now consider the decompositions (kerx kerx and ranx (ranx. Then we have the following matrix representations: ( ( ( T1 S1 S 2 X1 T, S, X, (2 T 2 T 3 S 3 where T1 is p-hyponormal, S 1 is injective (p,k-quasihyponormal and X 1 is injective with dense range. Therefore we have X 1 T 1 x XTx SXx S 1 X 1 x for x (kerx. (21 That is, X 1 T 1 S 1 X 1 and hence T 1 and S 1 are normal by Theorem 8 and X 1 T1 S 1 X 1 by the Fuglede-Putnam theorem. Therefore by Lemma 1, (kerx and ranx reduces T and S, respectively. Hence we obtain the XT S X. In Lemma 1, we can drop the injective condition if T is p-hyponormal instead of (p,k-quasihyponormality (see [7, Lemma 2]. Therefore we recapture a generalized Fuglede-Putnam theorem for p-hyponormal operators. Corollary 12. Let T ( is a p-hyponormal operator and let S ( is a p- hyponormal operator. If XT SX for X (,, then XT S X. Acknowledgment This work was supported by a Grant (R from the Korea Science and Engineering Foundation. References [1] A. Aluthge, On p-hyponormal operators for <p<1, Integral Equations and Operator Theory 13 (199, no. 3, [2] S. C. AroraandP. Arora,On p-quasihyponormal operators for <P<1, Yokohama Mathematical Journal 41 (1993, no. 1, [3] S. L. Campbell and B. C. Gupta, On k-quasihyponormal operators, Mathematica Japonica 23 (1978/1979, no. 2, [4] M. Chō and M. Itoh, Putnam s inequality for p-hyponormal operators, Proceedings of the American Mathematical Society 123 (1995, no. 8, [5] B.P. Duggal, On p-quasihyponormal operators for <p<1, Yokohama Mathematical Journal 41 (1993, [6] B. C. Gupta and P. B. Ramanujan, Onk-quasihyponormal operators II, The Tohoku Mathematical Journal 2 (1968, [7] I.H.JeonandB.P.Duggal,p-hyponormal operators and quasisimilarity, Integral Equations and Operator Theory 49 (24, no. 3, [8] I. H. Kim, On (p, k-quasihyponormal operators, Mathematical Inequalities & Applications 7 (24, no. 4, [9] I. H. Sheth, On hyponormal operators, Proceedings of the American Mathematical Society 17 (1966,

7 In Hyoun Kim 7 [1] J. G. Stampfli and B. L. Wadhwa, An asymmetric Putnam-Fuglede theorem for dominant operators, Indiana University Mathematics Journal 25 (1976, no. 4, [11] K. Tanahashi, A. Uchiyama, and M. Chō, Isolated points of spectrum of (p, k-quasihyponormal operators, Linear Algebra and its Applications 382 (24, [12] A. Uchiyama, An example of a p-quasihyponormal operato, Yokohama Mathematical Journal 46 (1999, no. 2, [13] J. P. Williams, Operators similar to their adjoints, Proceedings of the American Mathematical Society 2 (1969, In Hyoun Kim: Department of Mathematics, Seoul National University, Seoul , Korea address: ihkim@math.skku.ac.kr