Ralph Howard* Anton R. Schep** University of South Carolina

Transcription

1 NORMS OF POSITIVE OPERATORS ON L -SPACES Ralh Howard* Anton R. Sche** University of South Carolina Abstract. Let T : L (,ν) L (, µ) be a ositive linear oerator and let T, denote its oerator norm. In this aer a method is given to comute T, exactly or to bound T, from above. As an alication the exact norm V, of the Volterra oerator Vf(x)= R x f(t)dt is comuted.. Introduction For < let L [, ] denote the Banach sace of (euivalence classes of) Lebesgue measurable functions on [,] with the usual norm f =( f dt). For a air, with, < and a continuous linear oerator T : L [, ] L [, ] the oerator norm is defined as usual by (-) T, =su{ Tf : f =}. Define the Volterra oerator V : L [, ] L [, ] by (-2) Vf(x)= x f(t)dt. The urose of this note is to show that for a class of linear oerators T between L saces which are ositive (i.e. f a.e. imlies Tf a.e.) the roblem of comuting the exact value of T, can be reduced to showing that a certain nonlinear functional euation has a nonnegative solution. We shall illustrate this by comuting the value of V, for V defined by (-2) above. We first state this result. If << then let denote the conjugate exonent of, i.e. = so that + =. Forα, β > let be the Beta function. B(α, β) = t α ( t) β dt 99 Mathematics Subject Classification. 47A3, 47B38, 47G5. Key words and hrases. Oeratornorms, Positive linear oerator, Volterra oerator. *Research suorted in art by the National Science Foundation under grant number DMS , **Research suorted in art by a Research and Productive Scholarshi grant from the University of South Carolina

2 2 RALPH HOWARD* ANTON R. SCHEP** Theorem. If <, < then the norm V, V : L [, ] L [, ] is of the Volterra oerator (-3) V, =( ) (+ ) B(, ) In the case = this reduces to (-4) V, = ( ) B(, ) Secial cases of this theorem are known. When = =2kis an even integer, then the result is euivalent to the differential ineuality of section 7.6 of [H-L-P]. This seems to be the only case stated in the literature. The cases that or euals or are elementary. It is easy to see that V, = V, =for and. It is also straightforward for < and < that V, =( + ) and V, =( + ). The roof of theorem is based on a general result about comact ositive oerators between L saces. This theorem in turn will be deduced from a general result about norm attaining linear oerators between smooth Banach saces (see section 2 for the exact statement of the result). In what follows (, µ)and(,ν) will be σ-finite measure saces. If T : L (,ν) L (, µ) is a continuous linear oerator we denote by T the adjoint oerator T : L (, µ) L (,ν). For any real number x let sgn(x) be the sign of x (i.e. sgn(x) =forx>, = forx< and = for x = ). Then for any bounded linear oerator T : L (,ν) L (, µ) with<,< we call a function f L (, µ) acritical oint of T if for some real number λ we have (-5) T (sgn(tf) Tf )=λsgn(f) f (such function f is at least formally a solution to the Euler-Lagrange euation for the variational roblem imlicit in the definition of T, ). In the case that T is ositive and f a.e. (-5) takes on the simler form (-6) T ((Tf) )=λf For future reference we remark that the value of λ in (-5) and (-6) is not invariant under rescaling of f. If f is relaced by cf for some c>thenλis rescaled to c λ. Recall that a bounded linear oerator T : between Banach saces is called norm attaining if for some f we have Tf = T f. In this case T is said to attain its norm at f. The following theorem will be roved in section 2. Theorem 2. Let <,< and let T : L (,ν) L (, µ) be a bounded oerator. (A) If T attains its norm at f L (, µ), thenfis a critical oint of T (and so satisfies (-5) for some real λ).

3 NORMS OF POSITIVE OPERATORS ON L -SPACES 3 (B) If T is ositive and comact, then (-6) has nonzero solutions. If also any two nonnegative critical oints f,f 2 of T differ by a ositive multile, then the norm T, is given by (-7) T, = λ f where f is any nonnegative solution to (-6) In section 2 we give an extension of theorem 2(A) to norm attaining oerators between Banach saces with smooth unit sheres and use this result to rove theorem 2B. Theorem 2 is closely related to results of Graślewicz [Gr], who shows that if T is ositive, and (-6) has a solution f>a.e. forλ=,then T, =. In section 4 of this aer we indicate an extension of this result. We rove that if there exists a <f a.e. such that (-8) T (Tf) λf, then T, λ in case = and in case <we have T, λ Tf under the additional hyothesis that Tf L. Ineuality (-8) can be used to rove a classical ineuality of Hardy. Another alication of this result is a factorization theorem of Maurey about ositive linear oerators from L into L. It is worthwhile remarking that in case = = 2 the euation (-5) reduces to the linear euation T Tf = λf. In this case theorem 2 is closely related to the fact that in a Hilbert sace the norm of a comact oerator is the suare root of the largest eigenvalue of T T. 2. Norm attaining linear oerators between smooth Banach saces. Let E be a Banach sace and let E denote its dual sace. If f E then we denote by f (f) =<f,f >the value of f at f E. If f Ethen f E norms f if f = and <f,f >= f. By the Hahn-Banach theorem there always exist such norming linear functionals. A Banach sace E is called smooth if for every f E there exists a uniue f E which norms f. Geometrically this is euivalent with the statement that at each oint f of the unit shere of E there is a uniue suorting hyerlane. It is well known that E is smooth if and only if the norm is Gâteaux differentiable at all oints f E (see e.g. [B]). If E is a smooth Banach sace and f E, thendenotebyθ E (f) the uniue element of E that norms f, note Θ E (f) =. For the basic roerties of smooth Banach saces and the continuity roerties of the ma f Θ E (f) we refer to [B, art 3 Chater ]. The basic examles of smooth Banach saces are the saces L (, µ) where <<. For f L (, µ) one can easily show that (2-) Θ L (f) = f ( ) sgn(f) f by considering when euality holds in Hölder s ineuality. The following roosition generalizes art (A) of theorem 2 to norm attaining oerators between smooth Banach saces.

4 4 RALPH HOWARD* ANTON R. SCHEP** Proosition. Let T : E F be a bounded linear oerator between smooth Banach saces. If T attains its norm at f E then there exists a real number α such that (2-2) T (Θ F (Tf)) = αθ E (f) and the norm of T is given by (2-3) T = α Proof. Define Λ, Λ 2 E by Λ (h) =<h,θ E (f)> Λ 2 (h)= T <Th,Θ F(Tf) >= T <h,t (Θ F (Tf)) >. Then Λ =(since Θ E (f) =)andλ (f)= f,soλ norms f. Similarly Θ F (Tf) = imlies that Λ 2, but using Tf = T f we have Λ 2 (f)= f. Therefore Λ 2 also norms f. The smoothness of E now imlies that Λ =Λ 2. Hence (2-2) holds with α = T as claimed. Theorem 2(A) now follows from the following lemma. Lemma. If E = L (, µ),f = L (,ν) with <,< and f is a solution of (2-2), then f is a critical oint of f, i.e. where T (sgn(tf) Tf )=λsgn(f) f (2-4) λ = α f Proof. First we note that if fsatisfies (2-2), then we have Tf =<Tf,Θ F (Tf) >=<f,t Θ F (Tf)=<f,αΘ E (f)>=α f. Substitution of (2-) into (2-2) and multilication by Tf gives T (sgn(tf) Tf )=α Tf f ( ) This comletes the roof of the lemma and of theorem 2(A). sgn(f) f = α f sgn(f) f. To rove theorem 2(B), we first make the observation that if T : E F is a comact linear oerator and E is reflexive, then T attains its norm (since every bounded seuence in E contains a weakly convergent subseuence and T mas weakly convergent seuences onto norm convergent seuences). If now T is a ositive comact oerator from L (, µ) intol (,ν), then T attains its norm at a nonnegative f L (, µ) (simly relace f by f, ift attains its norm at f). If the additional hyothesis of theorem 2(B) holds, then any other nonnegative critical oint f is a ositive multile of f and therefore T also attains its norm at f.now the roosition and the lemma imly that T = α, whereαsatisfies (2-4). Hence (-7) holds. This comletes the roof of Theorem 2.

5 NORMS OF POSITIVE OPERATORS ON L -SPACES 5 3. The norm of the Volterra oerator In this section we shall rove Theorem. We first notice that the adjoint oerator of the Volterra oerator is given by (3-) V g(x) = x g(t)dt a.e. Since x f(t)dt and g(t)dt) are absolutely continuous functions, we can assume x that V (f),resectively V (g), eual these integrals everywhere. From Theorem 2(B) and the rescaling roerty of λ it follows that to rove Theorem it suffices to show that (3-2) V ((Vf) )=λf has a uniue ositive solution in L [, ] normalized so that (3-3) Vf() = f(t)dt = Since Vf is chosen to be absolutely continuous, we see that V ((Vf) )canbe chosen to be continuously differentiable on [, ]. Hence any nonnegative solution of (3-2) can be assumed to be continuously differentiable on [, ]. Also if f is a nonnegative solution of (3-2) normalized so that (3-3) holds, then Vf is nonnegative and Vf()=sothatVf is ositive on a neighborhood of x =. From (3-) we conclude that V ((Vf) ) is ositive on [, ). Hence any nonnegative solution of (3-) and (3-2) can be assumed to be strictly ositive and continuously differentiable on [, ). Assume now that f is such a solution of (3-) satisfying (3-2). Take the derivative on both sides in (3-) and then multily both sides by f to get the following differential euation (3-4) (Vf) f =λ( )f f. Using that f is the derivative of Vf, we can integrate both sides to get (3-5) (Vf) = λ( ) f since Vf() = and f() = by (3-2). To simlify the notation we let v(x) = Vf(x). Then v(x) > forx>, v (x) > forx<, v () = and (3-5) becomes (3-6) ( v(x) )= λ( ) v (x) or (3-7) c, = v (x) v(x)

6 6 RALPH HOWARD* ANTON R. SCHEP** where (3-8) c, =( λ( ) ). Using that v() = we can integrate (3-7) to get (3-9) c, x = v(x) t dt. Putting x = in this euation we get (3-) c, = dt = t B(, )= B(, ). The integral in (3-) was reduced to the Beta function by the change of variable t = u. The euations (3-9) and (3-) uniuely determine the function v and therefore also f = v and the number λ. This shows that (3-2) and (3-3) have a uniue nonnegative solution. Moreover starting with v and λ given by (3-9) and (3-) one sees by working backwards that f = v is a nonnegative solution of (3-2) and (3-3). Therefore by Theorem 2(B) the norm of V is given by (-7). From euations (3-) and (3-8) we can solve for λ to obtain (3-) λ ( ) = B(,. ) In case = this shows that V, = λ, which roves (-4). In case we need to comute v = f. To do this, multily (3-7) by v, raise the result to the ower, and then multily by v to obtain (3-2) v (x) = c, v (x)( v(x) ). Using that v() = and v() = we can integrate (3-2) to obtain (3-3) f = v = c, ( t ) dt = c, B(, +) =c, = B(, ) ( + ). (Here we used the identity B(α, β +)= (3-4) f = + B(, ) β α+βb(α, β). ) Therefore B(, ) ( )( ) ( + ) Using (3-) and (3-4) in formula (-7) now gives formula (-3) and the roof of theorem is comlete.

7 NORMS OF POSITIVE OPERATORS ON L -SPACES 7 4. Bounds for norms of ositive oerators In this section we shall consider a ositive oerator T acting on a sace of (euivalence classes of) measurable functions and give a necessary and sufficient condition for T to define a bounded linear oerator from L (,ν) intol (, µ) where < < and obtain a bound for T,, similar to (-7). Let L (, µ) denote the sace of a.e. finite measurable functions on and let M(, µ) denote the sace of extended real valued measurable functions on. For some alications it is useful to assume that T is not already defined on all of L. Therefore we shall assume that T is defined on an ideal L of measurable functions, i.e. a linear subsace of L (,ν) such that if f L and g f in L,theng L. By L + we denote the collection of nonnegative functions in L. A ositive linear oerator T : L L (, µ) is called order continuous if f n f a.e. and f n,f L imly that Tf n Tf a.e.. We first rove that such oerators have adjoints. Lemma. Let L be an ideal of measurable functions on (,ν) and let T be a ositive order continuous oerator from L into L (, µ). Then there exists an oerator T t : L (, µ) + M(,ν) + such that for all f L + and all g L (, µ) + we have (4-) (Tf)gdµ = f(t t g)dν. Proof. Assume first that there exists a function f > a.e. inl.letg L (, µ) +. Then we define φ : L + [, ] byφ(f)= (Tf)gdµ. SinceTf < a.e. we can find 2... such that for all n wehave (Tf )gdµ <. n Let L f = {h : h cf for some constant c} and define φ n : L f R by φ n (h) = (Th)gdµ. n The order continuity of T now imlies (through an alication of the Radon Nikodym theorem) that there exists a function g n L (,f dν) such that for all h L f we have φ n (h) = hg n dν, see e.g. [Z,theorem 86.3]. Moreover we can assume that g g 2... a.e.. Let g =sug n. An alication of the monotone convergence theorem now gives (Th)gdµ = hg dν for all h L f. The order continuity of T and another alication of the monotone convergence theorem now give (4-2) (Tf)gdµ = fg dν

8 8 RALPH HOWARD* ANTON R. SCHEP** for all f L. If we ut T t g = g, then (4-2) imlies that (4-) holds in case L contains a strictly ositive f. In case no such f exists in L, then we can find via Zorn s lemma a maximal disjoint system (f n )inl + and aly the above argument to the restriction of T to the functions f L with suort in the suort n of f n. We obtain that way functions g n with suort in n so that for all such f we have (Tf)gdµ = fg n dν n Now define T t g =sug n and one can easily verify that in this case again (4-) holds. This comletes the roof of the lemma. The above lemma allows us to define for any ositive oerator T : L L (, µ) an adjoint oerator T. Let N = {g L (, µ) :T t ( g ) L (,ν)} and define T g = T t g + T t g for g N. It is easy to see that T is ositive linear oerator from N into L (,ν) such that (4-3) (Tf)gdµ = f(t g)dν holds for all f L and g N. Observe that in case T : L L is a bounded linear oerator and, < then T as defined as above is an extension of the Banach sace adjoint. The above construction is motivated by the following examle. Examle. Let T (x, y) beµ ν-measurable function on. Let L = {f L (,ν) such that T (x, y) f(y) dν < a.e.} and define T as the integral oerator Tf(x)= T(x, y)f(y)dν(y) onl. Then one can check (using Tonelli s theorem) that N = {g L (, µ) such that T (x, y) g(x) dµ < a.e.} and that the oerator T as defined above is the the integral oerator T (x, y)g(x)dµ(x). We now resent a Hölder tye ineuality for ositive linear oerators. The result is known in ergodic theory (see [K], Lemma 7.4). We include the short roof. Abstract Hölder ineuality. Let L be an ideal of measurable functions on (,ν) and let T be a ositive oerator from L into L (, µ). If<< and =, then we have (4-4) T (fg) T(f ) T(g ) for all f,g with fg L, f L and g L. Proof. For any two ositive real numbers x and y we have the ineuality x y x + y,sothatif f,g with fg L, f L and g L, then for any α> (4-5) T (fg)=t((αf)( α )g) T ((αf) )+ T(( α g) ) = α T (f )+ α T(g ) Now for each x such that T (f )(x) choose the number α so that α T (f )(x) = α T(g )(x). Then (4-5) reduces to (4-4).

9 NORMS OF POSITIVE OPERATORS ON L -SPACES 9 Theorem 3. Let L be an ideal of measurable functions on (,ν) and let T be a ositive order continuous linear oerator from L into L (, µ). Let< < and assume there exists f L with <f a.e. and there exists λ>such that (4-6) T (Tf ) λf and in case <also (4-7) Tf L (, µ). Then T can be extended to a ositive linear ma from L (,ν) into L (, µ) with (4-8) T, λ Tf in case <and in case = (4-9) T, λ. If also f L (,ν), then (4-) T, λ f. Proof. Define the ositive linear oerator S : L (,ν) L (, µ) by Sf = (Tf ) Tf, note that S = T in case =. Then it is straightforward to verify that S (h) =T ((Tf ) h). This imlies that S (Sf ) = S ((Tf ) ( ) )=T (Tf ) λf, i.e. S satisfies (4-6) with =. Let n = {y : L ( n,ν) L.Let u L ( n,ν). Then we have (Su) dµ = S(uf f ) dµ = u f + S (Sf ) ( ) dν u f + λf dν by (4-6) Hence = λ u. n f (y) n}. Then S(u f + )(Sf ) dµ (Abstract Hölder ineuality) (4-) Su λ u

10 RALPH HOWARD* ANTON R. SCHEP** for all u L ( n,dν). If u L, letu n =min(u, n)χ. Then u n n u a.e. and (4-) holds for each u n. The order continuity of T and the monotone convergence theorem imly that S, λ. Note that in case = this roves (4-9). In case <define the multilication oerator M, bymh =(Tf ) h. Then (4-7) imlies, by means of Hölder s ineuality with r =,r =, that M, Tf. The ineuality (4-8) follows now from the factorization T = MS. Ineuality (4-) follows from (4-8) by using the ineuality Tf T, f and solving for T,. This comletes the roof of the theorem. The above theorem is an abstract version of what is called the Schur test for boundedness of integral oerators (see [H-S] for the case = = 2 and see [G],Theorem.I for the case < < ). Corollary. Let L be an ideal of measurable functions on (,ν) and let T be a ositive order continuous linear oerator from L into L (, µ). Let< < and assume there exists f L (,ν) with <f a.e. and there exists λ>such that (4-2) T (Tf ) = λf. Then T can be extended to a ositive linear ma from L (,ν) into L (, µ) with (4-3) T, = λ Tf = λ f and T attains its norm at f. Proof. If we multily both sides of (4-2) by f andthenintegrate,weget (4-4) (Tf ) dµ = λ (f ) dν. This imlies that Tf L (, µ), so that by the above theorem the ineualities (4-8) and (4-) hold. Euality (4-4) shows that Tf = λ f,fromwhich it follows that T, λ f. Hence we have euality in (4-). From this it easily follows that (4-3) holds and that Tf = T, f. Remark. In the above corollary one could hoe that in case = the euation (4-2) without the hyothesis f L still would imly that T, = λ. Theorem 3 still gives ineuality (4-9), but this is all what can be said as can be seen from the following examle. Let = =[, )withµ=νeual to the Lebesgue measure and define the integral oerator T by Tf(x)= x f(t)dt. An easy comutation shows that for << the euality (4-2) holds for some constant λ = λ(α), whenever f (y) =y α for all <α<. One can verify that in this case α = gives the best uerbound for T,, in which case λ =( ). Ineuality (4-9) is then the classical Hardy ineuality. We now state a converse to the above theorem, which is essentially due to [G, Theorem.II]. For the sake of comleteness we suly a roof, which is a simlification of the roof given in [G]. x

11 NORMS OF POSITIVE OPERATORS ON L -SPACES Theorem 4. Let T : L (,ν) L (, µ) be a ositive linear oerator and assume <,<. Then for all λ with λ > T, there exists <f a.e. in L (,ν) such that (4-5) T (Tf ) λf. Proof. We can assume that T, =. Then we assume that λ>. Now define S : L (,ν) + L (,ν) + by means of Sf =(T (Tf) ). Then it is easy to verify that f imlies that Sf, also that f f 2 imlies that Sf Sf 2 and that f n f a.e. in L imlies that Sf n Sf a.e.. Let now <f a.e. in L (,ν) such that f λ λ. For n>we define f n = f + λ Sf n. By induction we verify easily that f n f n+ and that f n for all n. This imlies that there exists f in L such that f n f a.e. and f. Now Sf n Sf imlies that f = f + λ Sf. Hence Sf λf, which is euivalent to (4-5) and f f > a.e., so that f > a.e. andtheroof is comlete. We resent now an alication of the revious two theorems. The result is due to Maurey ([M] ). Corollary. Let T : L (,ν) L (, µ) a ositive linear oerator and assume <<<. Then there exists <ga.e. in L r (, µ) with r = such that g T : L (,ν) L (, µ). Proof. From the above theorem it follows that there exists f L (,ν) such that (4-6) and (4-7) hold. The factorization follows now from the roof of Theorem 3. We conclude with another alication of Theorem 3. An ideal L of measurable functions is called a Banach function sace if L is Banach sace such that g f in L imlies g f. Theorem 5. Let L be a Banach function sace and assume that T and T are ositive linear oerators from L into L. Then T defines a bounded linear oerator from L 2 into L 2. Proof. Let S = T T.ThenSis a ositive oerator from L into L, sosis continuous (see [Z] ). Let λ>r(s), where r(s) denotes the sectral radius of S. From the Neumann series of the resolvent oerator R(λ, S) =(λ S) one sees that for all <g Lwe have f = R(λ, S)g λ g>andsf λf, i.e. T (Tf ) λf so (4-6) holds with = = 2. The conclusion follows now from theorem 3. A result for integral oerators similar to the above theorem was roved in [S],by comletely different methods. Remark. With some minor modifications of the roofs one can show that Theorems 3 and 4 and their corollaries also hold in case <.

12 2 RALPH HOWARD* ANTON R. SCHEP** Acknowledgements. The roblem which motivated this aer (finding methods to comute and estimate norms of concrete oerators like the Volterra oerator) came u in a conversation between one of the authors and Ken arnall. We also would like to thank Stehen Dilworth for ointing out that an earlier version of the roosition in section 2 had a suerfluous assumtion. References [B] B. Beauzamy, Introduction to Banach saces and their geometry, North Holland, 982. [G] E. Gagliardo, On integral transformations with ositive kernel, Proc. A.M.S 6 (965), [Gr] R. Grzaślewics, On isometric domains of ositive oerators on L saces, Collo. Math LII (987), [H L P] G.H. Hardy, J.E. Littlewood and G. Pólya, Ineualities, Cambridge University ress, 959. [H S] P.R. Halmos and V.S. Sunder, Bounded Integral Oerators on L 2 Saces, Sringer Verlag, 978. [K] U. Krengel, Ergodic Theorems, De Gruyter, 985. [M] B. Maurey, Théorèmes de factorisation our les oérateurs linéaires à valeurs dans les esaces L,Astérisue (974). [S] V.S. Sunder, Absolutely bounded matrices, Indiana Univ. J. 27 (978), [Z] A.C. Zaanen, Riesz saces II, North Holland, 983. Deartment of Mathematics, University of South Carolina, Columbia, SC 2928