Australia Joural of Basic ad Applied Scieces, 5(6): 253-257, 211 ISSN 1991-8178 A Characterizatio of Compact Operators by Orthogoality Abdorreza Paahi, Mohamad Reza Farmai ad Azam Noorafa Zaai Departmet of Mathematics, Saveh Brach, Islamic Azad Uiversity, Saveh, Ira Abstract: I this paper we exted the usual otio of orthogoality to Baach spaces Also we establish a characterizatio of compact operators o Baach spaces that admit orthoormal Schauder bases Key words: Orthogoality, Compact operator, Baach space INTRODUCTION Throughout this paper K is the field of real or complex umbers, E is a Baach space over K ad orm =1 L N deoted by, ad = = s a fiite or ifiite sequece i E,where either N is a positive iteger ad L= {1,2,,N} or N = 4 ad L = {1,2,} For { x : J} [ x : J] is deoted by ( )J L, the closure of the spa of the set The otio of orthogoality goes a log way bac i time Usually this otio is associated with Hilbert spaces or, more geerally, ier product spaces Various extesios have bee itroduced through the decades Thus, for istace, x is orthogoal to y i E (a) i the sese of [1], if for every αk, x y x (b) i the sese of [6], if for every αk, x y xy (c) i the isosceles sese [4], if x y x y (d) i the Pythagorea sese [4], if 2 2 2 x y x y (e) i the sese of [8], if x y x y x y x y Oe of the atural ad simple properties of orthogoality i a Hilbert space H that oe would lie to hold true i a Baach space is that x is orthogoal to y i H if ad oly if Correspodig Author: Abdorreza Paahi, Departmet of Mathematics, Saveh Brach, Islamic Azad Uiversity, Saveh, Ira E-mail: Paahi53@gmailcom, Apaahi@iau-savehacir 253
Aust J Basic & Appl Sci, 5(6): 253-257, 211 x y x y 1 2, K, = 1 2 1 2, (1) Clearly, i ay Baach space, Eq (1) is equivalet to x y = x y,, K (2) Hece, we itroduce the followig defiitio: Defiitio 11: A fiite or ifiite sequece ax = a x, L L for each axe L L i a Baach space E is said to be orthogoal if If, i additio, x 1for all L, the ( x) L is said to be orthoormal We write x y if x is orthogoal to y It is clear from the defiitio that is orthogoal i E if ad oly if is orthogoal i [ x : L] L Note that Defiitio 11 is a extesio of the usual otio of orthogoality sice i a Hilbert space H,, x y i our sese, if ad oly if, xy, =, where deotes the ier product i H Theorem 11: (Saidi, 22) Give a sequece L (i) The sequece is orthogoal i E L i E, the followig are equivalet: ( b ) ( c ) b c (ii) For each pair of sequeces ad i K satisfyig for all L, coverges, if ad oly if bx cx L L L bx L L coverges ad, if both coverge, L cx L Remar 11: (Jichaum, 29 Siger, 1957) Ivertibility of A is a ecessary coditio for existig the above factorizatio Characterizatio of Compact Operators: Let L(F,E) deote the set of bouded liear operators from the ormed space F ito the Baach space E It is ow that if F ad E are Hilbert spaces, the TL(F,E) is compact if ad oly if T is the limit i L(F,E) of a sequece of fiite-ra operators (Coway, 1985) This gives a coveiet ad practical characterizatio of compact operators i Hilbert spaces We show here that the same characterizatio still holds true for ay Baach space E that admits a orthoormal Schauder basis ad ay ormed space F More precisely, we have (3) 254
Aust J Basic & Appl Sci, 5(6): 253-257, 211 Defiitio 21: A liear trasformatio T: H 6H is compact if T(ball H) has compact closure i H Remar 21: The set of compact operators from H ito H is deoted by L( H,H) ie L( H, H)={ : H H H is the Hilbert space ad φ is a liear fuctio Remar 22: Let H be the Hilbert space, B ( H, H) B( H, H) B ( H, H) L (, ) H K is a liear space ad { T } L ( H, K) ad is such that T T the T B ( H, H) B ( H, H) if A BHH (, ), B LHH (, ) the TA ad BT are belog to Remar 23: If T B( H, H), the followig statemets are equivalet T is compact T* is compact there is a sequece {T } of operators of a ra such that T T H Corolary 21: If TB(H,H), the cl (rat) is separable ad if {e } is a basis for cl (rat) ad P is the proectio of H oto V{ e :1 } the PT T Example 21: If ( X,, ) is a measurable space ad K L 2 ( X X,, ) the 2 ( Kf )( x)= ( x, y) f ( y) d( y) Theorem 21: Suppose that { e } =1 is a compact operator ad K is a orthoormal Schauder basis of the Baach space E ad that F is a ormed space For each positive iteger, let P be the proectio o P ( e )= e, =1 =1 =1 e E [ e :1 ] defied by The, a operator TL(F,E) is compact if ad oly if P ET coverges to T i L(F,E) Proof: The sufficiecy part follows from the fact that for every Baach space E ad every ormed space F, the limit i L(F,E) of a sequece of fiite-ra operators is a compact operator (Hirsch ad Lacombe, 1999) Now, suppose that TL(F,E) is compact For each positive iteger, let T, = P ET Note that sice 255
Aust J Basic & Appl Sci, 5(6): 253-257, 211 { } e is orthoormal, it follows by Theorem 11 that P L(F,E) ad =1 for all Clearly we have, =1 { e } =1 sice is a Schauder basis of E, limp ( y)= y, for each y E Let B be the closed uit ball i F Sice T is compact, it follows that K = cl(t(b)) is a compact subset of E We eed to show that lim sup T ( x) T( x) xb Suppose this is ot true The there exist ε>, a subsequece, ad a sequece i B such that T T >, P { T } { } for all (4) Sice K is compact, there exists a subsequece of { x }, say { }, such that the sequece coverges i K to some yk The we have, sice P =1for all, x x { T} T T P ( T) P ( y) T P ( y) T y T P ( y) { T} { P ( y)} Lettig, sice ad both coverge to y, we obtai that lim T T =, which cotradicts Re (4) As a corollary we have Corolary 22: If E is a Baach space that admits a orthoormal Schauder basis ad F is a ormed space, the a operator TL(F,E) is compact if ad oly if it is the limit i L(F,E) of a sequece of fiite-ra operators Coclusio: I this wor, we exted the usual otio of orthogoality to Baach spaces Also, we establish a characterizatio of compact operators o Baach spaces that admit orthoormal Schauder bases I (Hirsch ad Lacombe, 1999), it is proved that every compact operator o Hilbert spaces is a limit of a sequeces of fiitera operators I this paper, we exteded this famous theorem, o Baach spaces It is a guess of authors that this theorem ca be exteded o topological vector spaces It is a ope problem, ow ACKNOWLEDGEMENTS Authors are grateful to the Islamic Azad Uiversity, Saveh Brach, for fiacial support to carry out this wor REFERENCES Birhoff, G, 1935 Orthogoality i liear metric spaces, Due Math J, 1: 169-172 Coway, JB, 1985 A Course i Fuctioal Aalysis, Spriger-Verlag Hirsch, F, G Lacombe, 1999 Elemets of Fuctioal Aalysis, Spriger-Verlag James RC, 1945 Orthogoality i ormed liear spaces, Due Math J, 12: 291-32 Jichau, H, H Li, 29 Characterizig isomorphisms i terms of completely preservig ivertibility or spectrum, J Math Aal Appl, 359: 81-87 256
Aust J Basic & Appl Sci, 5(6): 253-257, 211 Roberts, BD, 1934 O the geometry of abstract vector spaces, Tohou Math J, 39: 42-59 Saidi, FB, 22 A Extesio of the Notio of Orthogoality to Baach Spaces, J of Mathematical Aalysis ad Applicatios, 267: 29-47 Siger, I, 1957 Ughiuri abstracte si fuctii trigoometrice spatii Baach, Bul Stiit Acad RPR Sect Stiit Mat Fiz, 9: 29-42 257