BESSEL- AND GRÜSS-TYPE INEQUALITIES IN INNER PRODUCT MODULES

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1 Proceedigs of the Ediburgh Mathematical Society , 3 36 c DOI:0.07/S Prited i the Uited Kigdom BESSEL- AND GRÜSS-TYPE INEQUALITIES IN INNER PRODUCT MODULES SENKA BANIĆ, DIJANA ILIŠEVIĆ AND SANJA VAROŠANEC Faculty of Civil Egieerig ad Architecture, Uiversity of Split, Matice hrvatske 5, 000 Split, Croatia seka.baic@gradst.hr Departmet of Mathematics, Uiversity of Zagreb, Bijeička 30, 0000 Zagreb, Croatia ilisevic@math.hr; varosas@math.hr Received July 005 Abstract I this paper we give Bessel- ad Grüss-type iequalities i a ier product module over a proper H -algebra or a C -algebra. Keywords: Bessel iequality; Grüss iequality; ier product H -module; ier product C -module 000 Mathematics subject classificatio: Primary 6L08; 6H5 Secodary 6CXX; 6D99. Itroductio ad prelimiaries It is well kow that i a ier product space H,, over the real or complex umber field K the Bessel iequality holds. Namely, if {e i } i I is a family of orthoormal vectors i H, the for ay x Hwe have x, e i x, x.. i I Furthermore, some results cocerig upper bouds for the expressio x, x x, e i x H. i I ad for the expressio related to the Grüss-type iequality x, y x, e i e i,y x, y H.3 i I have appeared i [5]. I this paper we give a aalogue of the Bessel iequality. ad we ivestigate expressios aalogous to. ad.3 i a ier product module over a proper H - algebra or a C -algebra. 3

2 S. Baić, D. Ilišević ad S. Varošaec Throughout this paper we deote by A a proper H -algebra or a C -algebra. As we kow, a proper H -algebra is a complex Baach algebra A, whose uderlyig Baach space is a Hilbert space A,, equipped with a ivolutio a a satisfyig ab, c = b, a c = a, cb for all a, b, c A. A elemet a of a proper H - algebra A is called positive a 0 if ax, x 0 for every x A. Each a 0is self-adjoit a = a. For a proper H -algebra A, the trace class associated with A is τa ={ab : a, b A}. For every positive a τa there exists the square root of a, that is, a uique positive a / A such that a / = a. There are a positive liear fuctioal tr o τa ad a orm τ o τa, related to the orm of A by the equality tra a=τa a= a for every a A. A C -algebra is a complex Baach -algebra A, such that a a = a for all a A. A elemet a of a C -algebra A is called positive a 0 if it is self-adjoit ad has positive spectrum. The square root of a exists for every positive a A. I both structures, the relatio is give by a b if ad oly if b a 0. A ier product module over A is a right module H over A together with a geeralized ier product, i.e. with a mappig [, ] oh H whics τa-valued if A is a proper H -algebra or A-valued if A is a C -algebra satisfyig the followig properties: H [f,g + h] =[f,g]+[f,h] for all f,g,h H; H [f,ga] =[f,g]a for all f,g H, a A; H3 [f,g] =[g, f] for all f,g H; H [f,f] 0 for every f H; H5 [f,f] = 0 implies f =0. We shall say that H is a ier product H -module if A is a proper H -algebra ad that H is a ier product C -module if A is a C -algebra. A mappig [, ] satisfyig H H is called a geeralized semi-ier product, ad H is called a semi-ier product H -orc -module. The absolute value of f H is defied as the square root of [f,f] ad it is deoted by f. Let us emphasize that f is a positive elemet of A ad is thus self-adjoit. It is said that f,g H are orthogoal if ad oly if [f,g] =0. For some additioal facts about H -algebras, C -algebras ad semi-ier product modules over these structures we refer to the literature: see [8], [0], [], [], [3], [] ad the refereces therei.. Bessel-type iequalities A elemet p A is called a projectio if p is o-zero ad p = p = p.ifa is a C -algebra, the p =, while if A is a proper H -algebra, the p is ot equal to i geeral. A elemet h from a semi-ier product module over A is called a lifted projectio if h is a projectio i A. Accordig to [7, Lemma.], s a lifted projectio i a ier product module over A if ad oly if s o-zero ad h h = h.

3 Bessel- ad Grüss-type iequalities i ier product modules 5 The followig theorem gives us a A-valued Bessel-type iequality i a ier product module. Theorem.. Let A be a proper H -algebra or a C -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios. The, for all f H, [f, ][,f] f. holds. Proof. Usig the properties of the geeralized ier product ad the characterizatio of lifted projectios we have f [f, ][,f] =[f,f] =[f,f] =[f,f] =[f,f] =[f,f] = [ f [f, ][,f] [,f] [,f] [,f] [,f] [,f] [,f] [,f] [,f] [,f],f [f, ][,f]+ [f, ][,f]+ [f, ][,f]+ [f, ][,f]+ ] [,f] 0. [f, ][,f] [f, ][,f] [f, ] [,f] [,f] [,h j ][h j,f] i,j= The followig theorem gives a upper boud for the expressio f [f, ][,f], that is, the A-valued best approximatio formula. Theorem.. Let A be a proper H -algebra or a C -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios. For all f H ad a,...,a A, f [f, ][,f] f a i..

4 6 S. Baić, D. Ilišević ad S. Varošaec Proof. We have f a i f a i = [ f = f = f a i,f [,f] a i ] a i a i [,f] [f, ][,f]+ [f, ][,f]. [f, ] a i [,f] a i [f, ]a i + a i a i a i [,f]+ [f, ] a i a i a i Remark.3. Results such as these ca be foud i [] ad [], where a cocept of a orthoormal basis for Hilbert H -modules ad Hilbert C -modules is discussed. Let us emphasize that there are ier product H -modules ad ier product C - modules that lack the property of beig a complex vector space that is compatible with the structure of A see, for example, [6, Example.]. However, for λ C, h H ad a A we put λha :=hλa. I particular, if h H is a lifted projectio, the h h = h, so we are able to defie λh := hλ h ad we have λha = hλa =λha for every λ C ad every a A. Corollary.. Let A be a proper H -algebra or a C -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios. For all f H ad λ,...,λ C, f [f, ][,f] f λ i. Proof. It is sufficiet to defie a i = λ i ad apply Theorem.. 3. Grüss-type iequalities i ier product H -modules Let us recall the origial Grüss iequality [9]. Let f ad g be real fuctios that are defied ad itegrable o [a, b] R. If there exist real costats ϕ, φ, γ, Γ such that ϕ fx φ, γ gx Γ x [a, b],

5 Bessel- ad Grüss-type iequalities i ier product modules 7 the b a b a fxgxdx b a b a fxdx b a gxdx φ ϕγ γ. The costat Recetly, some papers o the Grüss iequality i real ad complex ier product spaces have appeared [3,, 5]. Here we reproduce the basic result from []. Theorem 3.. Let H,, be a ier product space over K K = R or C ad let e H, e =.Ifϕ, φ, γ, Γ are real or complex umbers ad x, y are vectors i H such that the coditios hold, the we have the iequality Re φe x, x ϕe 0 ad Re Γe y, y γe 0 3. x, y x, e e, y φ ϕ Γ γ. It is easy to see that the assumptios 3. are equivalet to the followig coditios: x φ + ϕ e φ ϕ ad y Γ + γ e Γ γ. Geeralizatios of Theorem 3. for ier product modules are give i [7]. I the followig text we give Grüss-type iequalities which are based o Bessel-type iequalities. ad. described i the previous sectio. Let H, [, ] be a ier product H -module or a ier product C -module ad let {h,...,h } be a set of mutually orthogoal lifted projectios i H. Below we will use the otatio [[, ]] for the mappig defied o H H by [[f,g]]=[f,g] [f, ][,g]. The mappig [[, ]] is a geeralized semi-ier product o H. The properties H H3 ca be verified by simple calculatio ad H is, i fact, the iequality.. Theorem 3.. Let A, be a proper H -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios i H. Ifa i,b i,c i,d i A i =,..., ad f,g H are such that the assumptios f ai + b i / a i b i, g ci + d i / c i d i 3.

6 8 S. Baić, D. Ilišević ad S. Varošaec hold, the we have the iequality τ [f,g] [f, ][,g] / a i b i c i d i a i b i f ai + b i / c i d i g ci + d i /. 3.3 The costat Proof. From the strog Cauchy Schwarz iequality see [0] for a semi-ier product H -module H, [[, ]], we get τ [f,g] [f, ][,g] = τ[[f,g]] tr[[f,f]] tr[[g, g]] =tr [f,f] [f, ][,f] tr [g, g] [g, ][,g] = = tr f f tr[,f] [,f] tr g [,f] g From Theorem. we have f [f, ][,f] f ai + b i, whicmplies that tr f [f, ][,f] tr f tr[,g] [,g] [,g]. 3. ai + b i. This yields f [,f] f ai + b i. 3.5

7 Bessel- ad Grüss-type iequalities i ier product modules 9 I the same way we obtai g [,g] g ci + d i. 3.6 Multiplyig 3.5 by 3.6 we get f Let us defie [,f] g [,g] f ai + b i g ci + d i. 3.7 m = / a i b i, p = / c i d i. Takig ito accout the assumptios 3., we ca also defie = q = a i b i f c i d i g ai + b i /, ci + d i /. Applyig the iequality m p q mp q,weget f ai + b i g ci + d i / a i b i c i d i a i b i f ai + b i / c i d i g ci + d i /. 3.8 The iequality 3.3 is obtaied after comparig the iequalities 3., 3.7 ad 3.8. If the submodule of H geerated by {h,...,h } is ot equal to H, the there exists h H such that h [,h]. If we put k = h [,h], the k H is o-zero ad for ay j {,,...,} we have [k, h j ]=[h, h j ] [h, ][,h j ]=[h, h j ] [h, h j ] h j =0.

8 30 S. Baić, D. Ilišević ad S. Varošaec Let us defie where f = kλ k + λ = ai + b i, g = kµ k + ci + d i, / k a i b i /, µ = k c i d i. For these f,g H, 3. ad 3.3 become equalities. More precisely, f ai + b i = / a i b i, g ci + d i = / c i d i so the secod summad o the right-had side i 3.3 vaishes, ad τ [f,g] [f, ][,g] = a i b i c i d i /. Hece, the costat Corollary 3.3. Let A, be a proper H -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios i H. Ifa i,b i,c i,d i A i =,...,, ad f,g H are such that the assumptios 3. hold, the τ [f,g] [f, ][,g] a i b i c i d i /. The costat For ϕ i,φ i,γ i,γ i C i =,...,weseta i = φ i,b i = ϕ i, c i = Γ i, d i = γ i i =,..., i Theorem 3. ad Corollary 3.3 to obtai the followig corollary. Corollary 3.. Let A, be a proper H -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios i H. Ifϕ i,φ i,γ i,γ i C i =,..., ad f,g H are such that the assumptios f g φ i + ϕ i Γ i + γ i / φ i ϕ i, / Γ i γ i 3.9

9 Bessel- ad Grüss-type iequalities i ier product modules 3 hold, the we have the followig iequality: τ [f,g] [f, ][,g] / φ i ϕ i Γ i γ i φ i ϕ i f φ i + ϕ i / Γ i γ i g / φ i ϕ i Γ i γ i. Γ i + γ i / The costat For =, Theorem 3. ad Corollary 3.3 become the geeralized Grüss iequality ad its refiemet. Corollary 3.5. Let A, be a proper H -algebra. Let H, [, ] be a ier product module over A ad let h H be a lifted projectio. If a, b, c, d A ad f,g H are such that the assumptios a + b f h hold, the we have the followig iequality: a b, g h c + d τ[f,g] [f,h][h, g] a b c d a b a + b / f h c d c d 3.0 c + d / g h a b c d. 3. The costat Furthermore, from Corollary 3. i the case i which = we obtai [7, Theorem.] ad [7, Corollary.]. Remark 3.6. For f = g, Corollary 3.3 becomes the Bessel-type iequality. Namely, we have the followig result. If the assumptios of Corollary 3.3 are satisfied ad if f ai + b i / a i b i 3.

10 3 S. Baić, D. Ilišević ad S. Varošaec holds, the we have the followig iequality: τ f [f, ][,f] a i b i.. Grüss-type iequalities i ier product C -modules Theorem.. Let A, be a C -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios i H. If a i,b i,c i,d i A i =,..., ad f,g H are such that the assumptios 3. hold, the we have the followig iequality: [f,g] [f, ][,g] / a i b i c i d i a i b i f c i d i g ai + b i / ci + d i /.. The costat Proof. First we have the Cauchy Schwarz iequality for a semi-ier product C - module H, [[, ]] see, for example, [8, Propositio.] together with [, Theorem..5.3]: [f,g] [f, ][,g] f [f, ][,f] g [g, ][,g].. Applyig [, Theorem..5.3] o., takig ito accout., we get f [f, ][,f] ai + b i f, that is, Aalogously, f g [f, ][,f] f [g, ][,g] g ai + b i..3 ci + d i..

11 Bessel- ad Grüss-type iequalities i ier product modules 33 Multiplyig.3 by. yields f [f, ][,f] g [g, ][,g] f ai + b i g ci + d i..5 The iequality 3.8 is the obtaied as i the proof of Theorem 3.. It remais to compare the iequalities.,.5 ad 3.8. The equality i. holds for the same f,g H that give the equality i 3.3. Note that the secod summad o the right-had side i. vaishes for such f ad g as well. Corollary.. Let A, be a C -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios i H. If a i,b i,c i,d i A i =,..., ad f,g H are such that the assumptios 3. hold, the [f,g] [f, ][,g] / a i b i c i d i. The costat As i the case of ier product H -modules, Theorem. ad Corollary. for a i = φ i, b i = ϕ i, i =,...,, c i = Γ i, d i = γ i, give the followig corollary. Note that, i cotrast with the H -case, here we have =. Corollary.3. Let A, be a C -algebra. Let H, [, ] be a ier product module over A ad let {h,...,h } be a set of mutually orthogoal lifted projectios i H. If ϕ i,φ i,γ i,γ i C i =,..., ad f,g H are such that the assumptios f g φ i + ϕ i / φ i ϕ i, Γ i + γ i / Γ i γ i.6

12 3 S. Baić, D. Ilišević ad S. Varošaec hold, the we have the followig iequality: [f,g] [f, ][,g] / φ i ϕ i Γ i γ i φ i ϕ i f φ i + ϕ i / Γ i γ i g / φ i ϕ i Γ i γ i. Γ i + γ i / The costat For = we get [7, Theorem 5.] ad [7, Corollary 5.]. The followig result is the geeralized Grüss iequality ad its refiemet ad it is obtaied from Theorem. ad Corollary. for =. Corollary.. Let A, be a C -algebra. Let H, [, ] be a ier product module over A ad let h H be a lifted projectio. If a, b, c, d A ad f,g H are such that the assumptios 3.0 hold, the we have the followig iequality: [f,g] [f,h][h, g] a b c d a b a + b / f h c d c + d / g h a b c d..7 The costat Remark.5. If we put f = g i Corollary., the we obtai the Bessel-type iequality. More precisely, if the assumptios of Corollary. ad 3. hold, the f [f, ][,f] a i b i. 5. Grüss-type iequalities i complex ier product spaces A complex ier product space is a ier product module over the complex umbers. If H,, is a complex ier product space, the the absolute value i H coicides with the orm i H iduced by,, that is, with the mappig defied by f = f,f for every f H. Therefore, e His a lifted projectio if ad oly if e =, ad {e,...,e } is a set of mutually orthogoal lifted projectios i H if ad oly if it is a orthoormal set i H that is, e i =i =,..., ad e i,e j = 0 for i j. The the previous results give us the followig theorem.

13 Bessel- ad Grüss-type iequalities i ier product modules 35 Theorem 5.. Let H,, be a complex ier product space ad let be the orm i H iduced by,. Let {e,...,e } be a orthoormal set i H. Ifϕ i,φ i,γ i,γ i C i =,..., ad f,g Hare such that the assumptios f g φ i + ϕ i e i Γ i + γ i e i / φ i ϕ i, / Γ i γ i 5. hold, the we have the followig iequality: f,g f,e i e i,g / φ i ϕ i Γ i γ i φ i ϕ i f φ i + ϕ i / e i Γ i γ i g Γ i + γ i / e i / φ i ϕ i Γ i γ i. 5. The costat Usig the fact that Re φ i e i f,f ϕ i e i = ad Re Γ i e i g, g γ i e i = φ i ϕ i f Γ i γ i g φ i + ϕ i e i Γ i + γ i e i, we get [5, Theorem 5]. A aalogue of iequality 5. i the case of a real ier product space is [5, Theorem ]. Refereces. D. Bakić ad B. Guljaš, Hilbert C -modules over C -algebras of compact operators, Acta Sci. Math. Szeged 68 00, M. Cabrera, J. Martíez ad A. Rodríguez, Hilbert modules revisited: orthoormal bases ad Hilbert Schmidt operators, Glasgow Math. J , P. Ceroe ad S. S. Dragomir, A refiemet of the Grüss iequality ad applicatios, RGMIA Res. Rep. Coll. 5 00, Article.. S. S. Dragomir, A geeralizatio of Grüss s iequality i ier product spaces ad applicatios, J. Math. Aalysis Applic , 7 8.

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