MODULI OF CONTINUITY FOR OPERATOR VALUED FUNCTIONS

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1 MODULI OF CONTINUITY FOR OPERATOR VALUED FUNCTIONS PETER MATHÉ AND SERGEI V. PEREVERZEV Abstrct. We shll study the modulus of continuity of nonnegtive functions f defined for non-negtive self-djoint opertors A, B in some Hilbert spce nd tking vlues there. More precisely, we estblish the vlidity of the following type of inequlities f(a f(b Cf( A B + C A B, where it is nturl to ssume, tht f is continuous nd f(0 = 0. Such inequlities re vlid for non-negtive opertor monotone functions, s well s for certin more generl clss of opertor vlued ones. We show, tht this clss of functions is rich enough to cover virtully ll exmples, for which similr inequlities hve been proven before, minly in the context of discretiztions of illposed opertor equtions. 1. Introduction Let us recll the notion of modulus of continuity of rel-vlued function f : [0, c] R, ω(f, δ := sup { f(s f(t, s, t [0, c], s t δ}. In the context of pproximtion theory, the behvior of ω(f, δ s δ 0 is importnt. Therefore we gree to sy tht the modulus is dominted by ϕ ner 0, if there re 0 < ε c nd C <, such tht (1 f(s f(t Cϕ( s t, provided s t ε. If f ws Lipschitz continuous with constnt C, then (2 f(s f(t C s t, 0 s, t <, such tht its modulus is dominted by t t. As we shll motivte lter, it is interesting to hve relxed versions of such inequlities Dte: Februry 18, Mthemtics Subject Clssifiction. 47A30, secondry: 46E40, 47A56. Key words nd phrses. opertor monotone functions, modulus of continuity. 1

2 2 PETER MATHÉ AND SERGEI V. PEREVERZEV for functions which re not Lipschitz continuous. Precisely, we sk, for which functions f does there exist constnt C <, such tht (3 f(s f(t Cf( s t, 0 s, t <? It is cler, tht we my nd do ssume, tht f(0 = 0. It is esy to see, tht (3 holds with C = 1, whenever f ws non-decresing nd concve. Proof. Concvity yields for 0 x < t < the inequlity f(x x/tf(t. Applying this for x := s nd x := t s, respectively, we obtin s t f(t f(s nd t s f(t f(t s, t such tht 0 f(t f(s t s f(t f(t s. t Therefore, sometimes such functions re clled moduli of continuity themselves, see e.g. [8, Sect. 6.1]. Combining both estimtes, we immeditely see, tht for functions f = f 1 + f 2, where f 1 ws non-negtive concve nd f 2 ws non-negtive Lipschitz continuous the following inequlity holds true. (4 f(s f(t f( s t + C s t, 0 s, t <. We note explicitly, tht for < the right hnd side in (4 cn further be bounded by (1 + C/f 1 (f( s t, since s result of concvity we hve for 0 < x < tht f 1 (x xf 1 (/. Thus we my rephrse inequlity (4 by sying, tht the modulus of continuity of the sum of concve function nd Lipschitz prt is dominted by the concve prt ner 0. Below we shll investigte nlogs of inequlities like (4 for functions mpping non-negtive opertors in some Hilbert spce to non-negtive opertors. We identify, using the spectrl clculus for self-djoint opertors in Hilbert spces, functions on R with their extensions to functions of self-djoint opertors. First note, tht the introduction of the modulus of continuity for opertor vlued functions is stright forwrdly given by ω(f, δ := sup { f(a f(b, A, B c, A B δ}, where A nd B re non-negtive self-djoint opertors in some Hilbert spce. For Lipschitz continuous opertor vlued functions, which re defined through estimtes (5 f(a f(b C A B,

3 MODULI OF CONTINUITY FOR OPERATOR VALUED FUNCTIONS 3 the modulus of continuity is gin dominted by t t. We will extend such estimtes to functions which re not Lipschitz continuous on right neighborhood of 0. Opertor vlued inequlities like (4 hve been estblished in the context of discretiztions of ill-posed opertor equtions for two clsses of functions. (1 f(s = s µ, s > 0, for 0 < µ < 1, nd (2 f(s = log p (1/s, 0 < s < 1, for ny p > 0. In both cses, the proof ws bsed on n integrl representtion for power functions, in prticulr on [9, Chpt. 4,(14.2]. For further pplictions in numericl nlysis it is importnt to extend the pplicbility nd to revel the nture of such estimtes. It will turn out, tht the vlidity of estimtes like (4 in closely connected to the opertor vlued nlogs of monotonicity nd concvity. In our study we shll indicte, tht version of (4, precisely (9 below, holds true for wide clss of functions, in prticulr those, which re opertor monotone. In Section 3 we estblish some further permnence properties, which llow to extend our nlysis further, covering ll functions studied so fr. In Section 4 we conclude our study with some generliztion to the cse of opertors which re not self-djoint. We introduce the following notion of opertor monotone functions, nd refer to [3, 1] for detils. Definition 1. A function f : (0, R is opertor monotone, if for ny pir of self-djoint opertors A, B with spectr in (0, such tht A B, we hve f(a f(b. It follows from L owner s Theorem (see, e.g., [4, Sect. 2] tht ech opertor monotone function f dmits n integrl representtion s Pick function (6 f( = α + β + [ 1 λ λ ] λ µ(dλ, for some α 0, rel number β nd finite positive mesure µ on R, stisfying (λ µ(dλ <. If we now ssume f(0 = 0, then f is non-negtive nd we cn deduce β = (1/λ λ/(λ µ(dλ, which results in refined representtion (7 f( = α + (λ λ µ(dλ.

4 4 PETER MATHÉ AND SERGEI V. PEREVERZEV If we furthermore ssume monotonicity on some intervl (0,, for some given, then µ does not hve mss on (0,, see [3, Chpt. 2, 2, Lemm 1], we lso refer to [1, Prop. V.4.14]. We note, tht opertor monotone functions on (0, re opertor concve, see [5, Thm. 2.5]. For such functions the modulus of continuity is dominted by itself s shows the following Theorem 1 (see [1, Thm. X.1.1]. Let f be opertor concve on (0, with f(0 = 0. For ny pir A, B of non-negtive self-djoint opertors on the Hilbert spce Hwe hve (8 f(a f(b f( A B. This result serves s strting point for our investigtion. We lso mention the importnt Exmple 1 (L owner-heinz. The function f( := µ, > 0, is opertor-concve whenever 0 < µ < 1. Actully, the following representtion cn be derived from [9, Chpt. 4, eq. (14.2]. µ = sin(πµ π(µ (λ λ d(( λµ Functions monotone on some [0, We estblish the following generliztion of Theorem 1. Theorem 2. Fix > 0. Let f : [0, R + be opertor monotone on (0,, stisfying f(0 = 0. For ech 0 < c < there is constnt C <, such tht for ny pir of non-negtive self-djoint opertors A, B with A, B c we hve (9 f(a f(b f( A B + C A B. Proof. By the bove considertions we hve (10 f( = α + 0 (λ λ µ(dλ + (λ λ µ(dλ. The first two terms correspond to n opertor concve function, sy f 1, such tht we my pply Theorem 1 to get f(a f(b f 1 ( A B + A(λ I A 1 B(λ I B 1 µ(dλ λ. The second integrl corresponds to Lipschitz continuous prt. First, it is esy to verify tht (11 A(λ I A 1 B(λ I B 1 = λ (λ I A 1 (λ I B 1.

5 MODULI OF CONTINUITY FOR OPERATOR VALUED FUNCTIONS 5 Using (11 for λ > c we cn estimte this integrl s following A(λI A 1 B(λI B 1 µ(dλ λ = = A B (λi A 1 (λi B 1 µ(dλ (λi A 1 (A B(λI B 1 µ(dλ A B ( c 2 µ(dλ (λ c 2 µ(dλ λ Since obviously f 1 ( f(, 0, we obtin (9. C A B. Exmple 2. The function f( := log 1 (1/, 0 < < 1, is opertor monotone on (0, 1. This cn be seen, since log( is Pick function s well s 1/. Since compositions of Pick functions re such, our clim is estblished. It is non-negtive for 0 < < 1. We my further extend the pplicbility to the cse, studied previously in [6]. There n inequlity, similr to (9 ws derived. Exmple 3. The functions log p (1/, 0 < < 1, re nonnegtive nd opertor monotone for ny 0 < p < 1. This follows by composing the functions from Exmple 1 with the function from Exmple 2. We end this list of exmples with the following one, for which inequlities like (9 were not known before. Exmple 4. By composing the functions from the previous exmples it follows tht the function log p log( 1, [0, 1 is non-negtive nd opertor monotone for ny p (0, 1. Functions of such type nturlly rise in the context of the inverse diffrction problems (see, e.g., [2, Th. 3.1]. As could be seen in the course of the proof of Theorem 2, ny function f, opertor monotone on some (0,, nd such tht f(0 = 0, is the sum of n opertor concve function on (0, nd Lipschitz continuous one, when considered on some sub-intervl [0, c], 0 < c <. Therefore (9 extends in nturl wy to ny function, which cn be decomposed in similr wy. We formulte this s Corollry 1. Let f : [0, c] R + be the sum of non-negtive opertor concve (i.e., on (0, nd non-negtive Lipschitz continuous

6 6 PETER MATHÉ AND SERGEI V. PEREVERZEV opertor vlued function, stisfying f(0 = 0. Then there is constnt C <, such tht for ny pir of non-negtive self-djoint opertors A, B with A, B c we hve (12 f(a f(b f( A B + C A B. In the light of the sme rgument s provided in the discussion fter (4 we cn clim, tht for such functions the modulus of continuity is dominted by themselves ner 0, i.e. there re 0 < ε c nd C <, such tht (13 f(a f(b Cf( A B, provided A B ε. 3. Extensions It will be convenient to formlize this resoning further. Let us consider the clss F(ε, C of ll functions, continuous on [0, c], nondecresing with f(0 = 0, which obey (13 on [0, c] with fixed constnt C nd A B ε. It is evident tht this clss is convex. As will be shown next, it is closed in the topology of uniform convergence on [0, c]. Theorem 3. If sequence (f n n N of functions in F(ε, C is uniformly convergent to some f, then f F(ε, C. Proof. For δ > 0 there is N N, such tht f n (s f(s δ, if n N, hence f n (A f(a δ for non-negtive self-djoint opertors A with norm A c. For such N nd A B ε we obtin f(a f(b f(a f n (A + f n (A f n (B + f n (B f(b 2 sup f(s f n (s + Cf n ( A B 0 s c (2 + Cδ + Cf( A B. Letting δ 0, we cn ccomplish the proof. Corollry 2. If sequence (f n n N of functions in F(ε, C is decresing point-wise nd f(s := inf n N f n (s is continuous, then f F(ε, C. This is consequence of Dini s Theorem, since monotonicity implies uniform convergence, provided the limit ws continuous. An nlog sttement is true for n incresing sequence nd the mximum function. We ccomplish our discussion with the following Theorem 4. Let 0 < p <. The function f p (s := log p (1/s on [0, 1/e] belongs to F(ε, C for some 0 < ε < 1/e nd C.

7 MODULI OF CONTINUITY FOR OPERATOR VALUED FUNCTIONS 7 This is vrint of n estimte in [6, Lemm 9]. Our derivtion cptures some ides used in [6], but employs previous results in this pper. Proof. The cses 0 < p < 1 re cptured by Exmple 3. First, let p > 1 be n integer. As in [6] we emphsize, tht f p is ( p. the decresing limit of ϕ µ s µ 0 where ϕ µ (s := In the 1 s µ µ light of Corollry 2 it is enough to show tht ll ϕ µ, µ 1/2 belong to F(ε, C for some common vlues 0 < ε < 1/e nd C. Uniformly in s 1/e we hve ( 1 s µ p ( p 1 + j = µ p µ p 1 j=0 = µ ( p 1 + j p p 1 j:µj 1 = f C (s + f H (s. s µj s µj + µ p j:µj>1 ( p 1 + j p 1 s µj The function f C is opertor concve, thus belongs to F(1/e, 1. We clim tht f H is H older continuous opertor-vlued with exponent 1/2, uniformly for µ. Indeed, for j let k := µj 2. Then 1 µj k 1 1. Moreover, k k 2 for non-negtive opertors A, B 1/e A µj B µj = (A k µj/k (B k µj/k A k B k µj/k ke µj/2 A B µj/k 2µje µj/2 A B 1/2 4e µj/4 A B 1/2, since µj/2 < e µj/4, µj 2. Therefore, f H (A f H (B 4µ p A B 1/2 j:µj>1 ( p 1 + j p 1 (e 1/4 µj 4f H (e 1/4 A B 1/2 4(5 p A B 1/2, providing uniform bound for ll ϕ µ, µ 1/2. Thus we rrive t f p (A f p (B f p ( A B + 4(5 p A B 1/2. Since f p is concve for 0 < t < e p 1, it is esy to check, tht crudely t 1/2 p p f p (t, there. Thus f p (A f p (B (1 + 4(5p p f p ( A B, which yields f p F(e p 1, C p with C p = 1 + 4(5p p. For non-integer vlues of p we could either refine the bove nlysis or, simpler, reduce

8 8 PETER MATHÉ AND SERGEI V. PEREVERZEV this to n with n 1 < p < n, to rrive t f p (A f p (B = f n (A p/n f n (B p/n f n (A f n (B p/n (C n f n ( A B p/n C n f p ( A B, which yields f p F(e p 1, C n, for non-integer vlues of p < n. Note tht the theorem just proved shows tht the clsses F(ε, C re much wider thn the clsses of opertor monotone functions. Indeed, one cn esily check tht for p > 1 the function f p is not Pick function on ny intervl (0, c, becuse its nlytic continution in the corresponding strip of upper hlf-plne hs imginry prt which is not lwys positive. Thus, f p is not n opertor monotone function. Nevertheless, for this function the inequlity (13 is still vlid. 4. Opertors which re not self-djoint Our previous results extend esily to the cse of opertors which re not self-djoint nd generlize the nlysis in [10, Chpt. 4, 1.3]. Given opertors A nd B we let A := (A A 1/2 nd B := (B B 1/2, respectively. We estblish Theorem 5. Let f F(ε, C be concve in right neighborhood of 0 (s rel vlued function. Then there is constnt C such tht ( (14 f( A f( B C 1 f A B log, A B s A B 0. Proof. By definition of F(ε, C we hve (15 f( A f( B Cf( A B, for A B ε. Now we use the following inequlity, estblished by T. Kto in [7], we refer to [1, Thm. X.2.5] for recent reference, (16 A B 2 ( A + B 2 + log A B. π A B (It ws lso shown, tht the logrithmic term cnnot be omitted in generl. If A B is smll enough there is D 1, such tht A B D A B log 1 A B.

9 MODULI OF CONTINUITY FOR OPERATOR VALUED FUNCTIONS 9 Inserting this into (15 we rrive t f( A f( B Cf(D A B log 1 A B, for some constnt C, gin for A B smll. Since f is concve for smll vlues of s > 0, we hve f(ds Df(s, which llows to complete the proof with C := D C. For µ < 1 nd f(s := s µ, in [10, Chpt. 4]. 0 < s, such result ws obtined References [1] Rjendr Bhti. Mtrix nlysis. Springer-Verlg, New York, [2] G. Bruckner, G. Elschner, nd M. Ymmoto. An optimiztion method for grting profile reconstruction. WIAS preprint 682, [3] Willim F. Donoghue, Jr. Monotone mtrix functions nd nlytic continution. Springer-Verlg, New York, [4] Frnk Hnsen. Opertor inequlities ssocited with Jensen s inequlity, in Survey on Clssicl Inequlities (T.M. Rssis, Ed., 67-98, Kluwer Acdemic Publishers Group, Dordrecht, [5] Frnk Hnsen nd Gert Kjergȧrd Pedersen. Jensen s inequlity for opertors nd L owner s theorem. Mth. Ann., 258(3: , 1981/82. [6] Thorsten Hohge. Regulriztion of exponentilly ill-posed problems. Numer. Funct. Anl. Optim., 21(3-4: , [7] Tosio Kto. Continuity of the mp S S for liner opertors. Proc. Jpn Acd., 49: , [8] N.P. Korneichuk. Exct Constnts in Approximtion Theory. Encyclopedi of Mthemtics & its Appl. Cmbridge University Press, Cmbridge, [9] M. A. Krsnosel skiĭ, P. P. Zbreĭko, E. I. Pustyl nik, nd P. E. Sobolevskiĭ. Integrl opertors in spces of summble functions. Noordhoff Interntionl Publishing, Leiden, [10] G. M. Vĭnikko nd A. Yu. Veretennikov. Itercionnye procedury v nekorrektnyh zdqh. Nuk, Moscow, Weierstr s Institute for Applied Anlysis nd Stochstics, Mohrenstr se 39, D Berlin, Germny E-mil ddress: mthe@wis-berlin.de of Mthemtics, Tereshen- Ukrinin Acdemy of Sciences, Inst. kivsk Str. 3, Kiev 4, Ukrine E-mil ddress: serg-p@mil.kr.net