COMPACTNESS AND BEREZIN SYMBOLS

Transcription

1 COMPACTNESS AND BEREZIN SYMBOLS I CHALENDAR, E FRICAIN, M GÜRDAL, AND M KARAEV Abstract We answer a question raised by Nordgren and Rosenthal about the Schatten-von Neumann class membershi of oerators in standard reroducing kernel Hilbert H saces in terms of their Berezin symbols 1 Introduction Let Ω be a subset of a toological sace X such that the boundary Ω is non-emty Let H be an infinite dimensional Hilbert sace of functions defined on Ω We say that H is a reroducing kernel Hilbert sace if the following two conditions are satisfied: (i) for any λ Ω, the functionals f f(λ) are continuous on H; (ii) for any λ Ω, there exists f λ H such that f λ (λ) 0 According to the Riesz reresentation theorem, the assumtion (i) imlies that, for any λ Ω, there exists k λ H such that f(λ) = f, k λ H, f H The function k λ is called the reroducing kernel of H at oint λ Note that by (ii), we surely have k λ 0 and we denote by ˆk λ the normalized reroducing kernel, that is ˆk λ = k λ / k λ H Following the definition of [NR94], we say that a reroducing kernel Hilbert sace H is standard if ˆk λ 0 (weakly) as λ ζ, for any oint ζ Ω In [NR94], E Nordgren and P Rosenthal established a characterization of comact oerators acting on such saces in terms of the Berezin symbols of their unitary orbits Recall that if L(H) denotes the sace of linear and bounded oerators on H, then the Berezin symbol T of any oerator T L(H) is the function defined on 2010 Mathematics Subject Classification Primary 47B38, 4B07; Secondary 47B35 Key words and hrases Berezin symbols, comact oerators, Schatten-von Neumann classes, reroducing kernel Hilbert sace, model saces This work was suorted by the PHC Boshore

2 2 I CHALENDAR, E FRICAIN, M GÜRDAL, AND M KARAEV Ω by T (λ) = T ˆk λ, ˆk λ H, λ H Then the characterization of Nordgren and Rosenthal is the following Theorem 11 [NR94, Corollary 28] Let H be a standard reroducing kernel Hilbert sace on Ω and let A L(H) Then A is comact if and only if lim λ ζ U 1 AU(λ) = 0, for every unitary oerator U on H and every oint ζ in Ω In [NR94], Nordgren and Rosenthal addressed two questions: Question 1: What can we say about comactness in reroducing kernel Hilbert saces which are not standard? Question 2: Is it ossible to characterize the Schatten-von Neumann class oerators in terms of their Berezin symbols? The aim of this note is to give answers to these two questions We also rovide a characterization of bounded oerators acting on reroducing kernel Hilbert saces in terms of the Berezin symbols of their unitary orbits We finally discuss the articular case where H is a backward shift invariant subsace K Θ associated with an inner function Θ 2 Berezin symbols, boundedness and comactness in reroducing kernel Hilbert saces Regarding Theorem 11, it is natural to ask if one can also characterize the boundedness of an oerator in terms of its Berezin symbols Contrary to the comactness question, the analogous boundedness roblem is trivial In fact, note that (21) U 1 AU(λ) = U 1 AU ˆk λ, k λ = AU ˆk λ U ˆk λ Hence, if A is bounded, we have su{ U 1 AU(λ) : λ Ω, U L(H) unitary } < Conversely, assume that there exists λ Ω satisfying su{ U 1 AU(λ) : U L(H) unitary } < Given an arbitrary vector h of norm 1 in H, there exists a unitary oerator U on H such that U ˆk λ = h According to (21), we get su Ah, h <, h H, h =1

3 COMPACTNESS AND BEREZIN SYMBOLS 3 which gives that A is bounded Therefore, we obtain that the following assertions are equivalent: (i) The oerator A is bounded (ii) su{ U 1 AU(λ) : λ Ω, U L(H) unitary } < (iii) There exists λ Ω such that su{ U 1 AU(λ) : U L(H) unitary } < Note that, in the boundedness roblem, the hyothesis of standardness of the reroducing kernel Hilbert sace H is not required Therefore, Question 1 of Nordgren and Rosenthal aears naturally In this direction, we will show that the hyothesis of standardness of the Hilbert sace H in Theorem 11 can be highly weakened For this urose, for any reroducing kernel Hilbert sace H on Ω (not necessarily standard), denote by H Ω the subset of the boundary of Ω defined by H Ω = {ζ Ω : ˆk λ 0 whenever λ ζ} It is clear from the definitions that H is standard if and only if H Ω = Ω In the case where H Ω, one can obtain an analogue of Theorem 11 The roof will follow along the same lines and the key oint is the following result which goes back to Dixmier [Dix49] Lemma 21 Let (f n ) be a weakly null sequence of unit vectors and let (δ n ) be a sequence of ositive numbers Then there exists a subsequence (f nk ) k 1 of (f n ) and an orthonormal sequence (h k ) k 1 such that f nk h k < δ k for all k 1 Theorem 22 Let H be a reroducing kernel Hilbert sace on Ω such that H Ω and let A L(H) Then the following assertions are equivalent (i) A is comact; (ii) for every oint ζ H Ω and every unitary oerator U on H, we have lim λ ζ U 1 AU(λ) = 0; (iii) there exists a sequence (λ n ) of oints in Ω, converging to a oint ζ H Ω, such that for every unitary oerator U on H, we have lim U 1 AU(λ n ) = 0 n +

4 4 I CHALENDAR, E FRICAIN, M GÜRDAL, AND M KARAEV Proof The imlication (i) = (ii) follows easily from (21) and the imlication (ii) = (iii) is trivial Therefore the only imlication to be roved is (iii) = (i) So, let (λ n ) be a sequence of oints in Ω, converging to a oint ζ H Ω, such that lim n + U 1 AU(λ n ) = 0, for every unitary oerator U on H To show that A is comact, we use a characterization due to Anderson and Stamfli [AS71, Lemma 3] which says that A is comact if and only if, for any orthonormal sequence (e n ) n, Ae n, e n H 0, as n + Furthermore, since Ae n, e n H 0 if and only if Ae 2n, e 2n H 0 and Ae 2n+1, e 2n+1 H 0, we may assume that the orthonormal sequence is such that san(e n : n 1) has infinite codimension So let us fix such a sequence (e n ) By Lemma 21, there exists a subsequence (ˆk λnl ) l 1 of (k λn ) and an orthonormal sequence (h l ) l 1 in H such that lim ˆk λnl h l = 0 l + By taking a subsequence if necessary, we may suose that the orthogonal comlement of the subsace generated by (h n ) is infinite dimensional Since both sequences (h n ) and (e n ) generate subsaces of infinite codimension, we can construct a unitary oerator U on H such that Uh l = e l, l 1 If follows that Ae l, e l = AUh l, Uh l AU(h l ˆk λnl ), Uh l + AU ˆk λnl, U ˆk λnl + AU ˆk λnl, U(h l ˆk λnl ) U 1 AU(λ nl ) + 2 A ˆk λnl h l, and thus ( Ae l, e l ) l 1 converges to 0, as l + Note that we give at the end of the aer an examle showing that the assumtion H Ω is indeed essential for our results 3 Berezin symbols and ideals of comacts of oerators Let T L(H) and let T = (T T ) 1/2 be its modulus Then it is well known that if T is comact, then T is also comact and hence its sectrum is given by a sequence tending to zero (if it is infinite) Let s n = s n (T ), n 1, be the eigenvalues of T arranged decreasingly and counted with multilicities The numbers s n are called the singular numbers of T Now let = ( n ) be a nonincreasing sequence of

5 COMPACTNESS AND BEREZIN SYMBOLS 5 ositive real numbers and let 1 < Then we define the class S by S = {T L(H) : T comact and (s n ) l ()} and let ( ) 1/ T S = s n n It is an exercise to check that S is a two-sided closed ideal of L(H) Moreover, we have (31) s 1 (T ) = T 1/ 1 T S In the articular case where n = 1, for any n 1, then S corresonds to the so-called Schatten-von Neumann classes S and if = 1 and n = 1/n, n 1, then S 1 corresond to the Matsaev class In the sequel we use the following characterization of membershi to S This result (recisely the equivalence of (i) and (ii)) is a articular case of a more general result, true for any symmetrically normed ideal (see for instance [GK69, Theorem III42]) However, for the sake of comleteness, we give a direct roof which follows along the same lines as the classical case corresonding to the Schatten-von Neumann classes (see [Zhu07, Theorem 127]) Lemma 31 Let T be a comact oerator on a Hilbert sace H, 1 < + and let = ( n ) be a decreasing sequence of ositive real numbers The following assertions are equivalent: (i) The oerator T is in S ; (ii) For all orthonormal sequences (e n ) in H, we have T e n, e n n < + ; (iii) For all orthonormal sequences (e n ) in H with infinite codimension, we have T e n, e n n < + Moreover, if ( ) 1/ M (T ) = su T e n, e n n : (e n ) orthonormal sequence, then (32) 4 1 min(1, )M (T ) T S max(1, )M (T )

6 6 I CHALENDAR, E FRICAIN, M GÜRDAL, AND M KARAEV Proof The equivalence between (ii) and (iii) follows from the observation that the sequence ( T e n, e n ) is in l () if and only if both sequences ( T e 2n, e 2n ) and ( T e 2n+1, e 2n+1 ) are in l () Now let us rove the equivalence between (i) and (ii) and the estimates (32) First assume that T is self-adjoint Then there exists an orthonormal set (σ n ) of H such that, for all x H, we have T x = t n x, σ n σ n, where (t n ) is the sequence of eigenvalues of T We can of course assume that the sequence (t n ) is arranged so that its modulus is decreasing to 0 Then we have s n (T ) = t n and T S = t n n Since t n = T σ n, σ n, we get T S = T σ n, σ n n, whence (33) T S M (T ) On the other hand, for any orthonormal sequence (e n ) in H, we have: T e k, e k = t n e k, σ n σ n, e k = t n e k, σ n 2 Let q be the conjugate index of (q = if = 1) Since by Parseval s inequality e k, σ n 2 e k 2 = 1, we obtain using Hölder s inequality, T e k, e k t n e k, σ n 2/ e k, σ n 2/q ( ) 1/ t n e k, σ n 2

7 COMPACTNESS AND BEREZIN SYMBOLS 7 Therefore T e k, e k k k t n e k, σ n 2 k 1 k 1 = k t n e k, σ n 2 + k t n e k, σ n 2 k 1 n k+1 k 1 n k = k t n e k, σ n 2 + t n k e k, σ n 2 k 1 k n n k+1 Now using the decreasing of both sequences ( t n ) and ( n ), we deduce T e k, e k k k t k e k, σ n 2 + t n n e k, σ n 2 k 1 k 1 n k+1 that is 2 t n n, (34) M (T ) 2 1/ T S This roves the desired result when T is self-adjoint For the general case we may write T = T 1 + it 2, where T 1 = T +T 2 and T 2 = T T are comact self-adjoint oerators It is clear that T is 2i in S if and only both T 1 and T 2 are in S Moreover, observing that T e n, e n 2 = T 1 e n, e n 2 + T 2 e n, e n 2, and using a standard argument of convexity, we have min(1, ) T en, e n T 1 e n, e n + T 2 e n, e n max(1, ) T en, e n Hence, (35) min(1, )M (T ) ( M (T 1 ) + M (T 2 ) ) 1 max(1, )M (T ) Now using (34), we have for i = 1, 2, M (T i ) 2 T i S 2 T S, which gives the left inequality in (32) For the right inequality, note that the convexity of t t, imlies that k n (M (T 1 ) + M (T 2 )) ( M (T 1 ) + M (T 2 ) ) 1, and with (33) and (35), we get T S ( T 1 S + T 2 S ) max(1, )M (T ), which gives the desired inequality

8 8 I CHALENDAR, E FRICAIN, M GÜRDAL, AND M KARAEV Remark 32 Let ( ) 1/ M(T ) = su T e n, e n n where the suremum is taken over all orthonormal sequences (e n ) n of infinite codimension Since for any othonormal sequences (e n ) n, we have T e n, e n n = T e 2n, e 2n 2n + T e 2n+1, e 2n+1 2n+1, n=1 we get n=1 n=0 (36) M (T ) M (T ) 2 1/ M (T ) We then have the following characterization of membershi to S in terms of Berezin symbols Theorem 33 Let H be a reroducing kernel Hilbert sace on Ω such that H Ω, A be a comact oerator on H, = ( n ) be a decreasing sequence of ositive numbers and let 1 The following assertions are equivalent: (i) The oerator A is in S (ii) There exists a sequence (λ n ) n of oints in Ω converging to a oint ζ in H Ω such that for all unitary oerator U on H, the sequence ( U 1 AU(λ n )) n is in l () Moreover, if A B = su{ ( U 1 AU(λ n )) n l () : U unitary oerator on H}, then we have (37) c 1 () A S A B c 2 (, ) A S, where c 1 () = 1 2 min(1, ) and c2 (, ) = 4 1 max(1, )+2 1/ 1 ( + m=1 Proof Suose first that A is in S and let (λ n ) be any sequence which tends to ζ H Ω By definition of H Ω, the sequence of normalized reroducing kernels (ˆk λn ) converges weakly to 0 According to Lemma 21, there exist a subsequence (ˆk λnm ) m 1 of (ˆk λn ) and an orthonormal sequence (h m ) m 1 such that ˆk λnm h m 1/ m m 2 ) 1/ w m m 2

9 COMPACTNESS AND BEREZIN SYMBOLS 9 Let U be an arbitrary unitary oerator on H Using (21), we have U 1 AU(λ nm ) = AU ˆk λnm, U ˆk λnm AU(ˆk λnm h m ), U ˆk λnm + AUh m, Uh m + AUh m, U(ˆk λnm h m ) AUh m, Uh m + 2 A ˆk λnm h m AUh m, Uh m + 2 A 1/ m m 2 Since A S and since (Uh m ) m 1 is an orthonormal sequence, it follows from Lemma 31 that ( AUh m, Uh m ) m 1 is in l () Consequently, we get that ( U 1 AU(λ nm )) m 1 is in l () Moreover, using (31) and (32), we have ( U 1 AU(λ nm )) m l () max(1, ) A S + ( 2 1/ w m 1 A S m 2 m 1 = c 2 (, ) A S ) 1/ Conversely, assume that there exists a sequence (λ n ) which tends to ζ H Ω such that for all unitary oerator U on H, ( U 1 AU(λ n )) n l () By Lemma 31, A S if and only if ( Ae n, e n ) is in l (), for all orthonormal sequences (e n ) with infinite codimension Now fix such a sequence (e n ) and fix a real number r > 1 By Lemma 21, there exist a subsequence (ˆk λnm ) m 1 of (ˆk λn ) and an orthonormal sequence (h m ) m 1 in H such that ˆk λnm h m 1/ m m r By taking a subsequence if necessary, we may suose that the orthogonal comlement of the subsace generated by (h m ) m 1 is infinite dimensional Since both sequences (h n ) and (e n ) generate subsaces of infinite codimension, we can construct a unitary oerator U

10 10 I CHALENDAR, E FRICAIN, M GÜRDAL, AND M KARAEV on H such that Uh m = e m, m 1 If follows that Ae m, e m = AUh m, Uh m AU(h m ˆk λnm ), Uh m + AU ˆk λnm, U ˆk λnm + AU ˆk λnm, U(h m ˆk λnm ) U 1 AU(λ nm ) + 2 A ˆk λnm h m U 1 AU(λ nm ) + 2 A 1/ m, m r and thus ( Ae m, e m ) m 1 is in l () Moreover, we have ( + ) 1/ w m ( Ae m, e m ) m l () A B + 2 A m r Letting r + gives Using (32) and (36), we get whence m=1 ( Ae m, e m ) m l () A B A S max(1, )M (A) 2 max(1, )M (A), A S 2 max(1, ) A B Since for standard reroducing kernel Hilbert saces, we have H Ω = Ω, we immediately have the following corollary Corollary 34 Let H be standard reroducing kernel Hilbert sace on Ω, A be a comact oerator on H, = ( n ) be a decreasing sequence of ositive numbers and let 1 The following assertions are equivalent: (i) The oerator A is in S (ii) There exists a sequence (λ n ) n of oints in Ω converging to a oint ζ in Ω such that for all unitary oerator U on H, the sequence ( U 1 AU(λ n )) n is in l () In the articular case where n = 1, n 1, we also obtain the following corollary which gives an answer to Question 2 of Nordgren and Rosenthal Corollary 35 Let H be a standard reroducing kernel Hilbert sace, A L(H) and let 1 The following assertions are equivalent: (i) The oerator A is in the Schatten-von Neumann class S

11 COMPACTNESS AND BEREZIN SYMBOLS 11 (ii) There exists a sequence (λ n ) n of oints in Ω converging to a oint ζ in Ω such that for all unitary oerator U on H, the sequence ( U 1 AU(λ n )) n is in l Moreover, if A B = su{ ( U 1 AU(λ n )) n l : U unitary oerator on H}, then we have (38) c 1 () A S A B c 2 () A S, where ( c 1 () = 1 + ) 1/ 2 min(1, ) and c2 () = max(1, )+2 m 2 m=1 Remark 36 Note that if T is a comact oerator on a standard reroducing kernel Hilbert sace H and if (λ n ) n is any sequence of oints in Ω converging to a oint ζ in Ω then, for all unitary oerator U on H, the sequence ( U 1 AU(λ n )) n is converging to 0 Hence there exists a subsequence which is in l () But this subsequence deends on the unitary oerator U In fact, according to Corollary 34, if T is in S, then one can find a sequence (λ n ), which does not deend on the unitary oerator U, such that the sequence ( U 1 AU(λ n )) n is in l () 4 An examle : the model saces In this section, we will study the examle of closed invariant backward shift subsaces We determine the corresonding subset H Ω and deduce then results concerning comactness and membershi to S classes Let H 2 denote the Hardy sace of the oen unit disc D Then it is well known that H 2 is a reroducing kernel Hilbert sace whose reroducing kernel at the oint λ in D is given by k λ (z) = (1 λz) 1, z D Now it is easy to see that H 2 is a standard reroducing kernel Hilbert sace If we are interested in (closed) subsaces of H 2, there is no reason for these subsaces to be still standard (see [Kar] where this question is discussed in its full generality) One of the most famous subsaces of H 2 are the closed backward shift invariant subsaces K Θ Recall that given an inner function Θ (that is a bounded analytic function in D whose radials limits are of modulus one almost everywhere on T = D), we define K Θ by K Θ = H 2 ΘH 2 = H 2 ΘH 2 0,

12 12 I CHALENDAR, E FRICAIN, M GÜRDAL, AND M KARAEV where H 2 0 = zh 2 It is easy to show that K Θ is a reroducing kernel Hilbert sace whose reroducing kernel at oint λ D is given by kλ Θ (z) = 1 Θ(λ)Θ(z) 1 λz, (z D) In general, K Θ is not standard and we will rovide, in this section, a characterization in terms of Carathéodory set of Θ Recall that if Θ admits the factorization Θ(z) = ( ) λ n λ n z λ n 1 λ n z ex ζ + z T ζ z dµ(ζ), then its Carathéodory set E Θ [AC70] is defined by { E Θ = ζ T : 1 λ n 2 } ζ λ n + dµ(τ) 2 T ζ τ < + 2 Recall also that the following assertions are equivalent: (i) ζ E Θ (ii) For any function f K Θ, f(z) has a nontangential limit when z ζ nontangentially (iii) We have lim inf z ζ 1 Θ(z) 2 1 z 2 < + The equivalence between (i) and (ii) aears in [AC70] while the equivalence of (ii) and (iii) is contained in [Sar94] Note that Sarason treats in fact the more general situation of de Branges-Rovnyak saces (see also [FM08] for an equivalence of (i) and (ii) in this more general situation) Moreover, if ζ E Θ, then f f(ζ) is continuous on K Θ and there exists kζ Θ K Θ such that f(ζ) = f, k Θ ζ Note that kζ Θ 0, otherwise all functions in K Θ would have nontangential limit 0 at ζ and the function 1 Θ(0)Θ obviously does not It is also known [Sar94] that if ζ E Θ, then kλ Θ kθ ζ (in norm), as λ ζ nontangentially The following result characterizes the weakly null sequences of normalized reroducing kernels Lemma 41 Let Θ be an inner function and let ζ T Then the following assertions are equivalent: (i) k λ Θ 0 (weakly) as λ ζ

13 COMPACTNESS AND BEREZIN SYMBOLS 13 (ii) ζ T \ E Θ (iii) We have 1 Θ(z) 2 lim = + z ζ 1 z 2 Proof (i) = (ii): argue by absurd and assume that k λ Θ 0 (weakly) as λ ζ but ζ E Θ Using the Ahern Clark s result then kλ Θ kθ ζ (in norm), as λ ζ nontangentially That imlies that k Θ λ k Θ ζ (in norm), as λ ζ nontangentially Therefore k Θ ζ = 0, but since k ζ Θ = 1, we get a contradiction (ii) = (iii): Let ζ T \ E Θ Then by Carathéodory s theorem, we have 1 Θ(z) 2 lim inf = +, z ζ 1 z 2 which imlies obviously (iii) (iii) = (i): Assume that Take f K Θ H Then we have 1 Θ(z) 2 lim = + z ζ 1 z 2 f, k λ Θ = kθ λ 1 2 f(λ) kλ Θ 1 2 f ( ) But since kλ Θ 1 = 1 λ 2 1/2, 1 Θ(λ) we get that 2 lim f, k λ Θ = 0, (f λ ζ H K Θ ) To conclude it remains to note that H K Θ is dense in K Θ (indeed, the reroducing kernels k Θ λ belong to H K Θ ) We immediately obtain the following characterization of model saces which are standard Theorem 42 Let Θ be an inner function Then the sace K Θ is standard if and only if E Θ = Using Theorem 42 and Theorem 11, we get the next result Corollary 43 Let Θ be an inner function such that E Θ = and let A be an oerator on K Θ Then A is comact on K Θ if and only U 1 AU(λ) 0, as λ ζ, for any ζ T and for any unitary oerator U on K Θ

14 14 I CHALENDAR, E FRICAIN, M GÜRDAL, AND M KARAEV If we use the notation of Section 2, Lemma 41 shows that, for any inner function Θ, the set KΘ D, associated with Θ, is T\E Θ Thus one can reformulate in this context Theorem 22 and we get immediately the following Theorem 44 Let Θ be an inner function such that T \ E Θ Let A L(K Θ ) The following assertions are equivalent: (i) The oerator A is comact (ii) There exists a sequence (λ n ) n in D converging to ζ T \ E Θ such that for all unitary oerator U L(H), ( U 1 AU(λ n )) n converges to 0 Of course, we have also a formulation of Theorem 34 in the context of model saces K Θ Note that there exist model saces K Θ with the roerty that E Θ = T and then KΘ D = (a concrete examle is given for instance in [AC70]) Now let A be a comact oerator on such a sace and assume that A is also ositive and injective Then, whenever λ ζ nontangentially for some ζ T and U is any unitary oerator on K Θ, we have U 1 AU(λ) = U 1 AU k Θ λ, k Θ λ = A1/2 U k Θ λ 2 A 1/2 U k Θ ζ 2 0 This shows that the assumtion H Ω is indeed essential for the results in the aer Acknowledgment: The authors wish to thank the referee for valuable comments References [AS71] J H Anderson and J G Stamfli Commutators and comressions Israel J Math 10 (1971), [AC70] P R Ahern and D N Clark Radial limits and invariant subsaces Amer J Math, 92: , 1970 [Dix49] J Dixmier Étude sur les variétés et les oérateurs de Julia, avec quelques alications Bull Soc Math France, 77:11 101, 1949 [FSW72] P A Fillmore, J G Stamfli and J P Williams On the essential numerical range, the essential sectrum, and a roblem of Halmos Acta Sci Math (Szeged), 33 (1972), [FM08] E Fricain and J Mashreghi Boundary behavior of functions in the de Branges Rovnyak saces Comlex Anal Oer Theory 2 (2008), no 1, 87 97

15 COMPACTNESS AND BEREZIN SYMBOLS 15 [GK69] I C Gohberg and M G Krein Introduction to the theory of linear nonselfadjoint oerators Translated from the Russian by A Feinstein Translations of Mathematical Monograhs, vol 18, 1969 American Mathematical Society, Providence [Kar] MT Karaev Use of reroducing kernels and Berezin symbols technique in some questions of oerator theory Forum Math, DOI: /FORM (to aear) [NR94] E Nordgren and P Rosenthal Boundary values of Berezin symbols In Nonselfadjoint oerators and related toics (Beer Sheva, 1992), volume 73 of Oer Theory Adv Al, ages Birkhäuser, Basel, 1994 [Sar94] D Sarason Sub-Hardy Hilbert saces in the unit disk University of Arkansas Lecture Notes in the Mathematical Sciences, 10 John Wiley & Sons Inc, New York, 1994 A Wiley-Interscience Publication [Zhu07] K Zhu Oerator theory in function saces, volume 138 of Mathematical Surveys and Monograhs American Mathematical Society, Providence, RI, second edition, 2007 I Chalendar, Université de Lyon; Université Lyon 1; Institut Camille Jordan CNRS UMR 5208; 43, boulevard du 11 Novembre 1918, F Villeurbanne address: chalenda@mathuniv-lyon1fr E Fricain, Université de Lyon; Université Lyon 1; Institut Camille Jordan CNRS UMR 5208; 43, boulevard du 11 Novembre 1918, F Villeurbanne address: fricain@mathuniv-lyon1fr M Gürdal, Suleyman Demirel University, Deartment of mathematics, Isarta, Turkey address: gurdalmehmet@sduedutr M Karaev, Suleyman Demirel University, Isarta Vocational School, Isarta, Turkey address: mubariztadigoglu@sduedutr