NORMS OF POSITIVE OPERATORS ON Lp-SPACES Ralph Howard* Anton R. Schep** University of South Carolina p q ct. ≤ → Abstra Let 0 T : L (Y,ν) L (X,µ) be a positive linear operator and let kT kp,q denote its operator norm. In this paper a method is given to compute kT kp,q exactly or to bound kT kp,q from above. As an application the exact norm kV kp,q of R x the Volterra operator Vf(x)= 0 f(t)dt is computed. 1. Introduction ≤ ∞ p For 1 p< let L [0, 1] denote the Banach space of (equivalenceR classes of) 1 1 k k | |p p Lebesgue measurable functions on [0,1] with the usual norm f p =( 0 f dt) . For a pair p, q with 1 ≤ p, q < ∞ and a continuous linear operator T : Lp[0, 1] → Lq[0, 1] the operator norm is defined as usual by (1-1) kT kp,q =sup{kTfkq : kfkp =1}. Define the Volterra operator V : Lp[0, 1] → Lq[0, 1] by Z x (1-2) Vf(x)= f(t)dt. 0 The purpose of this note is to show that for a class of linear operators T between Lp spaces which are positive (i.e. f ≥ 0 a.e. implies Tf ≥ 0 a.e.) the problem of computing the exact value of kT kp,q can be reduced to showing that a certain nonlinear functional equation has a nonnegative solution. We shall illustrate this by computing the value of kV kp,q for V defined by (1-2) above. We first state this result. If 1 <p<∞then let p0 denote the conjugate exponent 0 p 1 1 of p, i.e. p = p−1 so that p + p0 =1.Forα, β > 0let Z 1 B(α, β)= tα−1(1 − t)β−1dt 0 be the Beta function. 1991 Mathematics Subject Classification. 47A30, 47B38, 47G05. Key words and phrases. Operatornorms, Positive linear operator, Volterra operator. *Research supported in part by the National Science Foundation under grant number DMS- 8803585, **Research supported in part by a Research and Productive Scholarship grant from the University of South Carolina 1 2 RALPH HOWARD* ANTON R. SCHEP** Theorem 1. If 1 <p,q<∞then the norm kV kp,q of the Volterra operator V : Lp[0, 1] → Lq[0, 1] is 1 q−p 0 1 0 0 1 1 −1 (1-3) kV k =(p)qqp (p+q) pq B( , ) p,q q p0 In the case p = q this reduces to 1 0 0 1 p p k k p (p ) (1-4) V p,p = 1 1 B( p , p0 ) Special cases of this theorem are known. When p = q =2kis an even integer, then the result is equivalent to the differential inequality of section 7.6 of [H-L-P]. This seems to be the only case stated in the literature. The cases that p or q equals 1or∞are elementary. It is easy to see that kV kp,∞ = kV k1,q =1for1≤p≤∞ and 1 ≤ q ≤∞. It is also straightforward for 1 <p≤∞and 1 ≤ q<∞that 1 1 0 k k 1 p k k 1 q V p,1 =(p0+1 ) and V ∞,q =(q+1 ) . The proof of theorem 1 is based on a general result about compact positive operators between Lp spaces. This theorem in turn will be deduced from a general result about norm attaining linear operators between smooth Banach spaces (see section 2 for the exact statement of the result). In what follows (X, µ)and(Y,ν) will be σ-finite measure spaces. If T : Lp(Y,ν) → Lq(X, µ) is a continuous linear operator we denote by T ∗ the adjoint operator 0 0 T ∗ : Lq (X, µ) → Lp (Y,ν). For any real number x let sgn(x) be the sign of x (i.e. sgn(x)=1forx>0, = −1forx<0 and = 0 for x = 0 ). Then for any bounded linear operator T : Lp(Y,ν) → Lq(X, µ)with1<p,q<∞we call a function 0 =6 f ∈ Lp(X, µ)acritical point of T if for some real number λ we have (1-5) T ∗(sgn(Tf)|Tf|q−1)=λsgn(f)|f|p−1 (such function f is at least formally a solution to the Euler-Lagrange equation for the variational problem implicit in the definition of kT kp,q). In the case that T is positive and f ≥ 0 a.e. (1-5) takes on the simpler form (1-6) T ∗((Tf)q−1)=λf p−1 For future reference we remark that the value of λ in (1-5) and (1-6) is not invariant under rescaling of f.Iffis replaced by cf for some c>0thenλis rescaled to cq−pλ. Recall that a bounded linear operator T : X → Y between Banach spaces is called norm attaining if for some 0 =6 f ∈ X we have kTfkY = kTkkfkX .In this case T is said to attain its norm at f. The following theorem will be proved in section 2. Theorem 2. Let 1 <p,q<∞and let T : Lp(Y,ν) → Lq(X, µ) be a bounded operator. (A) If T attains its norm at f ∈ Lp(X, µ),thenfis a critical point of T (and so satisfies (1-5) for some real λ). NORMS OF POSITIVE OPERATORS ON Lp-SPACES 3 (B) If T is positive and compact, then (1-6) has nonzero solutions. If also any two nonnegative critical points f1,f2 of T differ by a positive multiple, then the norm kT kp,q is given by p−q 1 q (1-7) kT kp,q = λ q kfkp where f =06 is any nonnegative solution to (1-6) In section 2 we give an extension of theorem 2(A) to norm attaining operators between Banach spaces with smooth unit spheres and use this result to prove theo- rem 2B. Theorem 2 is closely related to results of Gra´slewicz [Gr], who shows that if T is positive, p ≥ q and (1-6) has a solution f>0a.e.forλ=1,thenkTkp,q =1. In section 4 of this paper we indicate an extension of this result. We prove that if there exists a 0 <f a.e. such that (1-8) T ∗(Tf)q−1 ≤ λf p−1, q 1 1 − 1 p then kT kp,p ≤ λ p in case p = q and in case q<pwe have kT kp,q ≤ λ p kTfkq under the additional hypothesis that Tf ∈ Lq. Inequality (1-8) can be used to prove a classical inequality of Hardy. Another application of this result is a factorization theorem of Maurey about positive linear operators from Lp into Lq. It is worthwhile remarking that in case p = q = 2 the equation (1-5) reduces to the linear equation T ∗Tf = λf. In this case theorem 2 is closely related to the fact that in a Hilbert space the norm of a compact operator is the square root of the largest eigenvalue of T ∗T . 2. Norm attaining linear operators between smooth Banach spaces. Let E be a Banach space and let E∗ denote its dual space. If f ∗ ∈ E∗ then we denote by f ∗(f)=<f,f∗ >the value of f ∗ at f ∈ E.If0=6 f∈Ethen f ∗ ∈ E∗ norms f if kf ∗k = 1 and <f,f∗ >=kfk. By the Hahn-Banach theorem there always exist such norming linear functionals. A Banach space E is called smooth if for every 0 =6 f ∈ E there exists a unique f ∗ ∈ E∗ which norms f. Geometrically this is equivalent with the statement that at each point f of the unit sphere of E there is a unique supporting hyperplane. It is well known that E is smooth if and only if the norm is Gˆateaux differentiable at all points 0 =6 f ∈ E (see e.g. [B]). If E is a smooth Banach space and 0 =6 f ∈ E,thendenotebyΘE(f) the unique ∗ element of E that norms f, note kΘE (f)k = 1. For the basic properties of smooth Banach spaces and the continuity properties of the map f 7→ ΘE (f) we refer to [B, part 3 Chapter 1]. The basic examples of smooth Banach spaces are the spaces Lp(X, µ)where 1<p<∞. For 0 =6 f ∈ Lp(X, µ) one can easily show that k k−(p−1) | |p−1 (2-1) ΘLp (f)= f p sgn(f) f by considering when equality holds in H¨older’s inequality. The following proposition generalizes part (A) of theorem 2 to norm attaining operators between smooth Banach spaces. 4 RALPH HOWARD* ANTON R. SCHEP** Proposition. Let T : E → F be a bounded linear operator between smooth Banach spaces. If T attains its norm at 0 =6 f ∈ E then there exists a real number α such that ∗ (2-2) T (ΘF (Tf)) = αΘE (f) and the norm of T is given by (2-3) kT k = α ∗ Proof. Define Λ1, Λ2 ∈ E by Λ1(h)=<h,ΘE(f)> 1 1 Λ (h)= <Th,Θ (Tf) >= <h,T∗(Θ (Tf)) >. 2 kTk F kTk F Then kΛ1k =1(sincekΘE(f)k=1)andΛ1(f)=kfk,soΛ1 norms f. Similarly kΘF (Tf)k= 1 implies that kΛ2k≤1, but using kTfk = kTkkfk we have Λ2(f )= kfk. Therefore Λ2 also norms f. The smoothness of E now implies that Λ1 =Λ2. Hence (2-2) holds with α = kT k as claimed. Theorem 2(A) now follows from the following lemma.