Quasinormalty and subscalarity of class p-wa(s, t) operators

Transcription

1 Functional Analysis, Approximation Computation , Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: Quasinormalty subscalarity of class p-was, t operators K Tanahashi a, T Prasad b, A Uchiyama c a Department of Mathematics, Tohoku Pharmaceutical University, Sendai, , Japan b The Institute of Mathematical Sciences, CIT Campus, Chennai-6 113, India c Department of Mathematical Science, Faculty of Science, Yamagata University, Yamagata, , Japan Abstract Let T be a bounded linear operator on a complex Hilbert space H let T U T be the polar decomposition of T An operator T is called a class p-was, t operator if T t T s T t tp T tp T s T t T s sp T sp where < s, t < p 1 We investigate quasinormality subscalarity of class p-was, t operators Dedicated to the memory of Professor Takayuki Furuta with deep gratitude 1 Introduction Let BH denote the algebra of all bounded linear operators on a complex Hilbert space H Aluthge [1] introduced p-hyponormal operator T which is defined as T T p TT p where < p 1, proved interesting properties of p-hyponormal operators by using Furuta s inequality [8] If p 1, T is called hyponormal Hence p-hyponormality is a generalization of hyponormality It is known that p-hyponormal operators have many interesting properties as hyponormal operators, for example, Putnam s inequality, Fuglede-Putnam type theorem, Bishop s property β, Weyl s theorem polaroid property After this discovery, many authors are investigating new generalizations of hyponormal operator Let T BH T T T 1 By taking U T x Tx for x H Ux for x ker T, T has a unique polar decomposition T U T with condition ker U ker T We say that T U T is the polar decomposition of T in this paper The authors [14] introduced class p-was, t operator as follows: Definition 11 An operator T is called a class p-was, t operator if T t T s T t tp T tp 1 Mathematics Subject Classification Primary 47B Keywords Class A operator; Class p-was, t operator; quasinormal; subscalar Received: 17 Marh 17; 7 April 17 Communicated by Dragan S Djordjević addresses: tanahasi@tohoku-mpuacjp K Tanahashi, prasadvalapil@gmailcom T Prasad, uchiyama@scikjyamagata-uacjp A Uchiyama

2 K Tanahashi, TPrasad A Uchiyama / FAAC , T sp T s T t T s sp where < s, t < p 1 It is known that p-hyponormal operators log-hyponormal operators are class 1-wAs, t operators for any < s, t Class 1-wA1, 1 is called class A class 1-wA 1, 1 is called w-hyponormal [7, 9, 1, 15] Hence the class of p-was, t operators is a generalization of the class of A w-hyponormal operators C Yang J Yuan [16 18] studied a class of wfp, r, q operators T, ie, T T p T r 1 q T p+r q T p+r1 1 q T p T r T p 1 1 q where < p, < r, 1 q If we take small p 1 such that < p 1 p+r qr p 1 p+rq 1 pq, then T is a class p 1 -wap, r operator Hence the class of p 1 -wap, r opertors is a generalization of the class of wfp, r, q operators Aluthge transformation [1] is a good tool in operator theory I B Jung, E Ko C Pearcy [11] studied spectral properties of Aluthge transformation Definition 1 Let T U T be the polar decomposition of T BH Then generalized Aluthge transformation is defined by Ts, t T s U T t where < s, t Main Results It is known that an operator T is a class p-was, t operator if only if Ts, t tp T tp T sp Ts, t sp by [14] As a continuation of [14], we investigate quasinormality subscalarity of class p- was, t operators To prove main results, we need the following Lemma The proof is essentially due to C Yang J Yuan Proposition 34 of [18] For completeness, we prove the following Lemma Lemma 1 [18] If T is a class p-was, t operator < s s 1, < t t 1, < p 1 p 1, then T is a class p 1 -was 1, t 1 operator Proof Let T be class p-was, t Then T t T s T t tp T tp 1 T sp T s T t T s sp We prove that T is a p-was 1, t 1 operator Then T is a p 1 -was 1, t 1 operator by Lowner-Heinz s inequality Let A 1 T t T s T t tp B 1 T tp Since 1 implies A 1 B 1, we have B r r 1 Ap 1 B 1 1+r p +r B 1+r 1

3 K Tanahashi, TPrasad A Uchiyama / FAAC , for any r > p 1 by Furuta s inequality [8] Let Then tp+β t T β T s T β 1 T tp+β t β t, p s + t tp 1, r β t tp Hence we have for any < w tp + β t Let for β t Then T β T s T β w s+β T w f s β T s T β T s s s+β f s β { T s T β T s+β+w s s+β } s s+β+w { T s T β T β T s T β } w s s+β T β T s s+β+w { T s T β T w T β T s} { T s T β+w T s} s s+β+w f s β + w Hence f s β is decreasing for β t Then, by, T sp T s T t T s sp { f s t } p s s+β+w { f s t 1 }p sp T s T t 1 T s 1 Let A T sp B sp T s T t 1 T s 1 Then A 1+r 3 A r 3 Bp 3 for any r 3 p 3 1 by Furuta s inequality [8] Let r3 A 1+r 3 p 3 +r 3 p 3 s + t 1 sp 1, r 3 s 1 s sp Then Since we have T sp+s1 s sp+s T s 1 T t 1 T s 1 s 1 s 1 +t 1 sp + s 1 s s 1 p s 1 s1 p, T s 1p T s 1 T t 1 T s 1 s 1 p s 1 +t 1

4 K Tanahashi, TPrasad A Uchiyama / FAAC , Similarly, we have Hence T is a p-was 1, t 1 operator T t 1 T s 1 T t 1 t 1 p s 1 +t 1 T t 1p At first, we investigate quasinormality of class p-was, t operator Let T U T be the polar decomposition We say T is quasinormal if U T T U It is known that if T is a class As, t operator with < s, t Ts, t is quasinormal, then T is also quasinormal by [13] We extend this resullt as follows Theorem Let T U T be a class p-was, t operator with < s, t < p 1 If Ts, t T s U T t is quasinormal, then T is also quasinormal Hence T coincides with its Aluthge transform Ts, t T s U T t if s + t 1 Proof Since T is a class p-was, t operator, Ts, t rp T rp Ts, t rp 3 for all r, min{s, t}] Then Douglas s theorem [5] implies that Hence ran Ts, t rp ran T rp ran Ts, t rp [ran Ts, t ] [ran T ] [ran Ts, t ] [ran Ts, t] where [M] denotes the norm closure of M H Since ker T ker T s U T t ker Ts, t, we have Hence Put [ran T ] ker T ker Ts, t ker Ts, t [ran Ts, t ] [ran Ts, t ] [ran T ] Let Ts, t W Ts, t be the polar decomposition of Ts, t Then E : W W U U the orthogonal projection onto [ran T ] the orthogonal projection onto [ran Ts, t] WW : F Ts, t 1 X on H [ran Ts, t] ker Ts, t Then X is injective has a dense range Since W [ran Ts, t], we have W1 W W Since Ts, t is quasinormal, W commutes with Ts, t Hence Ts, t rp W W Ts, t rp W Ts, t rp W W T rp W W Ts, t rp W Ts, t rp Ts, t rp W Ts, t rp W W Ts, t rp W W T rp W

5 K Tanahashi, TPrasad A Uchiyama / FAAC , X rp Ts, t rp W Ts, t rp W Since WW WW Ts, t rp WW WW T rp WW 1, 4 implies that Ts, t rp T rp are of the forms Ts, t rp X rp X Y rp T rp rp Z rp 4 5 where Y, Z Since X is injective has a dense range [ran Ts, t ] [ran T ], we have [ran Y] [ran Z] [ran T ] [ran Ts, t] ker Ts, t ker T Since W commutes with Ts, t Ts, t 1, we have W1 W X Y W1 X W Y X Y W1 W XW1 XW So W 1 X XW 1 W Y XW, hence [ran W 1 ] [ran W ] are reducing subspaces of X Since W W Ts, t Ts, t, we have W W Ts, t 1 Ts, t 1 Then W 1 W 1X W 1 W Y X W W 1X W W Y Y Hence W 1 W 1 1, W W Y Y X k W 1 W 1X k W 1 Xk W 1, Y k W W Y k W Xk W U11 U for all k 1,, Put U 1 Then Ts, t T U 1 U s U T t W Ts, t implies X s U11 U 1 X t W1 W Z s U 1 U Z t X Y X s U 11 X t X s U 1 Z t W1 X Z s U 1 X t Z s U Z t W Y Then X s U 11 X t W 1 X X s W 1 X t, X s U 1 Z t W Y X W X s U 11 W 1 X t, X s U 1 Z t X t W

6 K Tanahashi, TPrasad A Uchiyama / FAAC , Since X is injective has a dense range, we have U 11 W 1 U 1 Z t X t W Hence U 11 U 11 W 1 W 1 1 Since U U is the orthogonal projection onto [ran T ] [ran Ts, t] U 1 + U U 1 U 1 U 11 U 1 + U 1 U U 1 U 11 + U U 1 U 1 U 1 + U U on H [ran Ts, t] ker Ts, t, we have U 1, U 1 U 11 U U 1 U 1 U 1 + U U Since U 1 Z t X t W, we have Z t Z t U 1 U 1Z t W Xt W Y t Since < rp t 1, we have Z rp Z t U 1 U 1Z t rp t W Xt W rp t Y rp Z rp by Lowner-Heinz s inequality 5 Hence Z t U 1 U 1Z t rp t Z rp Y rp, so Z Y Since Ts, t T Z t Z t U 1 U 1Z t Z t U 1 U 1Z t + Z t U U Z t Z t U 1 U 1 + U U Z t Z t, we have Z t U U Z t Z t U This implies that [ran U ] ker Z On the other h U U UU implies U 11 1 U U U 1 1 U 1 + U U Hence U U 1 U 1 + U U U Hence U 11 U U 1 U 11 U 1 U 1 + U U U 1 U 1 U 1 + U U U ran U [ran U 1 U 1 + U U ] Hence U Then U [ran U U] [ran Ts, t] [ran T ] [ran Ts, t] [ran Z] ran U ker Z [ran Z] {} U11 U 1 W1 U 1 ran U [ran Ts, t] [ran T ] rane

7 K Tanahashi, TPrasad A Uchiyama / FAAC , Hence EU U Since W commutes with Ts, t T T, we have T s W U T t W T T s U T t W Ts, t Ts, t Hence EW UE EWE EUE Since E U U W W [ran W] [ran Ts, t] [ran T ] ran E, we have EW W Then U UU U UE EUE EWE WE WW W W Thus U W Since W commutes with Ts, t, we have U commutes with T Therfore T is quasinormal Corollary 3 Let T U T be a class p-was, t operator with < s, t < p 1 If Ts, t T s U T t is normal, then T is also normal Proof T is quasinormal by the above theorem Hence Ts, t U T Ts, t T U Thus This implies that T T therefore T is normal T Ts, t Ts, t T Next, we investigate subscalarity of class p-was, t operator Let X be a complex Banach space U C be an open subset Let OU, X denote the Fréchet space of all analytic X-valued functions on U with the topology of uniform convergence on compact subsets of U Also, Let EU, X denote the Fréchet space of all infinitely differentiable X-valued functions on U with the topology of uniform convergence of all derivatives on compact subsets of U We say that T satisfy Bishop s property β if T z f n z in OU, X f n z in OU, X for every open set U C f n z OU, X E Albrecht J Eschmeier [] proved that T BX satisfies Beshop s property β if only if T is subdecomposable, ie, T is a restriction of a decomposable operator We say that T satisfy Eschmeier-Putinar-Bishop s property β ϵ if T z f n z in EU, X f n z in EU, X for every open set U C f n z EU, X J Eschmeier M Putinar [6] proved that T BX satisfies Eschmeier-Putinar-Bishop s property β ϵ if only if T is subscalar, ie, T is a restriction of a scalar operator Theorem 4 If T is p-was, t with < s + t 1 < p 1, then T satisfies Bishop s property β Eschmeier-Putinar-Bishop s property β ϵ Hence T is subscalar Proof We may assume s + t 1 by Lema 1 Then Ts, t is minsp,tp -hyponormal by [14] Hence Ts, t satisfies Bishop s property β Eschmeier-Putinar-Bishop s property β ϵ by [4, 1] Then T satisfies Bishop s property β Eschmeier-Putinar-Bishop s property β ϵ by Theorem 1 of [3]

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