M-Quasihyponormal Composition Operators. on Weighted Hardy Spaces

Transcription

1 It. Joural of Math. Aalysis, Vol., 8, o. 4, M-Quasihypoormal ompositio Operators o Weighted Hardy Spaces S. Paayappa Departmet of Mathematics, Govermet Arts ollege oimbatore , amil Nadu, Idia paayappa@gmail.com D. Sethilkumar Departmet of Mathematics, Sri Ramakrisha Egieerig ollege oimbatore 641, amil Nadu, Idia sethilsekumhari@gmail.com R. Moharaj Departmet of Mathematics, Sri Ramakrisha Polytechic ollege oimbatore 641, amil Nadu, Idia moharajsrptc@gmail.com Abstract If is a aalytic fuctio mappig the uit disk D ito itself, we defie the compositio operator o the space H () by f = f. I this paper, we ivestigate the relatioship betwee properties of the symbol ad the quasihypoormality of the operators ad. Mathematics subject classificatio: Primary 47B38, secodary 47B37, 47B35. Keywords: Weighted Hardy Space, M-quasihypoormal, Quasihypoormal, Hypoormal, Partial isometry, ompositio Operators

2 1164 S. Paayappa, D. Sethilkumar ad R. Moharaj 1. PRELIMINARIES series. Let f be a aalytic map o the ope uit disk D give by the aylor s f(z) = a + a 1 z + a z +.. Let = { } α = as. be a sequece of positive umbers with = 1 ad he set H () of formal complex power series f(z) = a z such that = f = = a < is a Hilbert space of fuctios aalytic i the uit disc with the ier product. <f, g> = a for f as above ad g (z) = b z. = b Let D be the ope uit disk i the complex plae ad let : D D be a aalytic self-map of the uit disk ad cosider the correspodig compositio operator actig o H (), i.e., (f) = f, f H () he operators are ot ecessarily defied o all of H (). hey are everywhere defied i some special cases: o the classical Hardy space H (the case whe = 1 for all ). See for example [7], ad o a geeral space H () if the fuctio is aalytic o some ope set cotaiig the closed uit disk havig supremum orm strictly smaller tha oe (see [11]). here are a lot of other kow properties of compositio operators, o the classical Hardy space H (See for example [1], [5] ad [7]), ad o more geeral space H () (see [3], [4], [8], [1] ad [11]). I [], owe s ad Kriete obtaied a ice correlatio betwee hypoormality of compositio operators o H ad the Dejoy-Wolff poit of the iduced map. I [9], Nia Zorboska obtaied some results o the hypoormality of compositio operators ad their adjoits. I this article, we are iterested i the M-quasihypoormality of compositio operators ad their adjoits.. A operator o a Hilbert space H is called M-quasihypoormal if there exists M> such that =

3 M-quasihypoormal compositio operators 1165 M () If M = 1, is said to be quasihypoormal. Furuta et al. [6], itroduced a ew class class A operators as follows. A operator belogs to class A if ad oly if ( ) ½ ad showed that this class is icluded i the class of paraormal operators. Let be a poit o the ope disk. Defie Z k (z) = = he the fuctio k is a poit evaluatio for H (). he k is i H () ad k. = = hus, K is a icreasig fuctio of. If f(z) = = herefore, z a the a f,k = = = f () f,k = f ( ) for all f ad k is kow as the poit evaluatio keral at. It ca be easily show that k = k () ad k = 1 (the fuctio idetically equal to 1). heorem.1: If is M-quasihypoormal the k M () Proof:

4 1166 S. Paayappa, D. Sethilkumar ad R. Moharaj is M-quasihypoormal. M f,f - ( ) f, f, for all f H (). M f, f - f, f M f, f - f, f M f f Let f = k, we have, M k k M k M k k () M. [9] heorem. : k k () A partial isometry compositio operator o H () is M-quasihypoormal the M 1. Proof : is M-quasihypoormal. f M f, M k( for all f H () ) + k for all k > (M k( ) + k ) M k + k M k + k M k + k M f k f + k f

5 M-quasihypoormal compositio operators 1167 Let f = k, we have, M M k k k + k k k k + k k k M k k + k M k k + k M k + k By elemetary properties of real quadratic form, we get, M 1. heorem.3: If is hypoormal compositio operator o H () ad M-quasihypoormal the M 1. is Proof : is M-quasihypoormal M f,f ( ) f, f, for all f H () M M f, f f, f f, f f, f M f f Let f = k, we have, M M k k k k M k k () () Sice is hypoormal, we have () =. [9] M k k

6 1168 S. Paayappa, D. Sethilkumar ad R. Moharaj M M k () k 1 M 1 k heorem.4: Proof : he If be quasihypoormal o the space H (), the () =. Let be quasihypoormal o H () ad f f for all f i H () ad if f = k be poit evaluatio at. k, we have, k = which implies, sice = 1, that () =. k = k () 1 = = = () k k = k = k = 1 heorem.5: If is of lass A o H () the () =. Proof : If is of lass A, implies ( ) ( ) ( )

7 M-quasihypoormal compositio operators 1169 ( ) f, f f, f for all f i H (). Let f = f, f f, f f k, we have, k k k () k () f k k k = k = 1 1, by theorem 4, which implies that () =. Refereces [1]. owe, ompositio Operators o H, J. Operator heory, 9(1983), [].owe ad.l.kriete, Subormality ad compositio operators o H, J. Fuct. Aal., 81(1988), [3] B. Macluer, ompositio Operators o S p, Housto J. of Math., 13(1987), [4] B.Macluer ad J. Shapiro, Agular Derivatives ad ompact ompositio Operators o the Hardy ad Bergma Spaces, aad J. Math., 38(1986), [5] E.A. Nordgre, ompositio Operators i Hilbert Space, I : Hilbert space operators, Lecture Notes i Math., 693, Spriger-Verlag (Berli, Heidelberg, New York, (1977), [6] S. Paayappa ad D. Sethilkumar, lass A ompositio Operators, Bull. al. Math. Soc., 96(1) (4), [7] H.J. Schwartz, ompositio Operators o H p, hesis U. of oledo, [8] J. Shapiro, ompact ompositio Operators o Spaces of Boudaryregular Holomorphic Fuctios, Proc. Amer. Math. Soc., 1(1987),

8 117 S. Paayappa, D. Sethilkumar ad R. Moharaj [9] N. Zorboska, Hypoormal ompositio Operators o Weighted Hardy spaces, Acta Sci. Math., 55(1991), [1] N. Zorboska, ompact ompositio Operators o some Weighted Hardy Spaces, J. Operator heory, (1989), [11] N. Zorboska, ompositio Operators iduced by Fuctios with Supremum Strictly Smaller tha Oe, Proc. Amer. Math. Soc., 16(1989), Received: March 5, 8