Concr. Oper. 2021; 8:48–59 Research Article Open Access Fernanda Botelho* and Richard J. Fleming The dual of the space of bounded operators on a Banach space https://doi.org/10.1515/conop-2020-0109 Received August 23, 2020; accepted January 6, 2021 Abstract: Given Banach spaces X and Y, we ask about the dual space of the L(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X** and Y*. Keywords: Tensor products; dual spaces; dual of spaces of operators MSC: Primary 46B78, secondary 46B10 Dedicated to the memory of our friend Professor James E. Jamison. 1 Introduction The space L(X, Y) of bounded linear operators from the Banach space X to the Banach space Y is one of the fundamental spaces of study in Banach space theory, and it seems natural to inquire about its dual space. A preliminary search of articles on this subject gave a rather sparse result. Hence we thought it might be of use to investigate an answer to this question. Let us begin with a very naive and elementary approach. In the case where Y is one dimensional, the answer is just X*, so what is the answer in the two dimensional case? It depends, of course, on the norm of the space. Let us consider Y = `∞(2), where X is any Banach space. ∞ Clearly T 2 L(X, ` (2)) can be written as Tx = (φ1(x), φ2(x)) for x 2 X and where φ1, φ2 are elements of * * X . Furthermore, kTk = maxfkφ1k, kφ2kg. It follows that if ψ 2 L(X, Y) , then ψ = (ψ1, ψ2) where ψ1, ψ2 2 ** 1 ** X and kψk = kψ1k + kψ2k. Thus, the dual space in this case is the space ` (2, X ). This can be extended inductively to any nite dimension. Corresponding results for other common norms can also be obtained. Notation is fairly standard. Because of a variety of uses of the letter B, we will use L(X, Y) to denote the bounded operators from X to Y and similarly, LA(X, Y) for the approximable operators (those for which the nite rank operators are dense), and LK(X, Y) for the compact operators from X to Y. The closed unit ball of a Banach space X is denoted by BX. In years gone by when researchers were discussing a particular result, someone would often say, “Well, it’s probably somewhere in Grothendieck!" That is the case here as well. A more elegant approach to the description of L(X, Y)* involves tensor products, and the result that L(X, `∞(n))* = `1(n, X**) which we de- ∞ * ** ∞ * scribed above could be expressed by L(X, ` (n)) = X ⊗ˆ π(` (n)) . Before showing how that works, let us take a little refresher course on tensor products. *Corresponding Author: Fernanda Botelho: Department of Mathematical Sciences, University Of Memphis, Memphis TN 38152; E-mail: [email protected] Richard J. Fleming: Department of Mathematics, Central Michigan University, Mt. Pleasant, Michigan 48859; E-mail: [email protected] Open Access. © 2020 FernandaBotelho and Richard J. Fleming, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. The dual of the space of bounded operators on a Banach space Ë 49 2 Tensor products We mention here some basic and useful facts about tensor products for the benet of readers who may not be familiar with the subject. What is reported here is material taken from the wonderful book of Raymond Ryan [7]. Let B(X × Y) denote the space of bilinear functions from X × Y, where X, Y are vector spaces, to the scalar eld K (which is always the real or complex numbers). In its most elementary form, the tensor product X ⊗ Y can be thought of as a subspace of the space of linear functionals on B(X × Y). Given x 2 X and y 2 Y, let x ⊗ y denote the linear functional on B(X × Y) dened by (x ⊗ y)(A) = hA, x ⊗ yi = A(x, y), where A is a bilinear form on X × Y. The tensor product X ⊗ Y is the space spanned by these elements. Thus an element of this tensor product can be written n X u = λi xi ⊗ yi i=1 and since such representations are not unique, we may always write an element in the form n X u = xi ⊗ yi . i=1 The projective norm, π, on the tensor product of two Banach spaces X ⊗ Y is dened by ( m m ) X X π(u) = inf kxikkyik : u = xi ⊗ yi . i=1 i=1 The inmum is taken, of course, over all representations of u. Then π is a norm with the property that π(x⊗y) = kxkkyk for x 2 X and y 2 Y. The tensor product X ⊗ Y with the projective norm is denoted by X ⊗π Y and its 1 1 completion is given by X⊗ˆ π Y. Some useful facts for future reference are that ` ⊗ˆ π Y = ` (Y) and this holds also for the space `1(J) where J is an arbitrary indexing set. An interesting and important result about the projective tensor product is given in the following theorem. We think it instructive to oer a proof, although we give an alternative to the interesting one given in [7]. Theorem 1. Let X and Y be Banach spaces, u 2 X⊗ˆ π Y, and ϵ > 0. Then there exist bounded sequences fx g fy g in X Y respectively, such that the series P∞ x ⊗ y converges to u and n , n , n=1 n n X∞ kxnkkynk < π(u) + ϵ. n=1 Proof. u 2 X⊗ Y ϵ u Pk1 x ⊗ y π u u ϵ π u Given ˆ π and > 0 there exists 1 = i=1 i i such that ( − 1) < /8 and ( 1) ≤ Pk1 kx k ky k π u ϵ π u π u ϵ Pk1 kx k ky k π u ϵ ϵ π u ϵ i=1 i i < ( 1) + /4. Thus ( 1) < ( ) + /8 so that i=1 i i < ( ) + /8 + /4 < ( ) + /2. Without loss of generality, we may assume that, for all i 2 f1, 2, . k1g, kxik = kyik. We assume this as well Pk2 xi yi u u 2 X⊗ˆ π Y u xi ⊗yi π u u u ϵ for all , chosen below. Since − 1 there exists 2 = i=k1+1 such that ( − 1 − 2) < /16 Pk2 π u kxik kyik π u ϵ π u π u u ϵ ϵ ϵ and ( 2) ≤ i=k1+1 < ( 2) + /8. Then ( 2) ≤ ( − 1) + /16 < /8 + /16, and therefore π(u2) < ϵ/4. n+2 We continue this process to dene a sequence fung such that π(u − u1 − u2 − · · · − un) < ϵ/2 , where Pkn Pkn n+1 un xi ⊗ yi π un kxik kyik π un ϵ π un π u u un = i=kn−1+1 , and ( ) ≤ i=kn−1+1 ≤ ( )+ /2 . Therefore ( ) ≤ ( − 1 −· · · −1)+ ϵ/2n+1 < ϵ/2n+2 + ϵ/2n+1 < ϵ/2n. It follows that X∞ X∞ u = un = xi ⊗ yi , n=1 i=1 50 Ë Fernanda Botelho and Richard J. Fleming and ∞ ∞ X X n kxik kyik < π(u) + ϵ/2 = π(u) + ϵ. i=1 n=1 p We observe that sequences fkxikg and fkyikg are bounded by π(u) + ϵ. This completes the proof. It follows from the above result that ( ) X∞ X∞ X∞ π(u) = inf kxnkkynk : kxnkkynk < ∞, u = xn ⊗ yn , n=1 n=1 n=1 where the inmum is taken over all such representations of u. This is another of several ways of calculating the projective norm. Dual spaces are of particular interest to us, so let us describe the dual space of the projective tensor product of two Banach spaces, X, Y. A bilinear form B on X × Y is bounded if there exists C > 0 such that jB(x, y)j ≤ Ckxkkyk for all x 2 X and y 2 Y. Let B(X, Y) denote the space of bounded bilinear forms on X × Y, which is a Banach space with the norm kBk = supfjB(x, y)j : x 2 BX , y 2 BY g. Given B 2 B(X × Y) there exists a unique linear functional B˜ on X⊗ˆ π Y satisfying B˜(x ⊗ y) = B(x, y), and it is straitforward to show that the correspondence B $ B˜ is an isometric isomorphism between B(X × Y) and * * (X⊗ˆ π Y) . We note also, that given B 2 B(X × Y), we can dene an operator LB : X ! Y by hy, LB(x)i = B(x, y) * and the correspondence B $ LB describes an isometric isomorphism between B(X × Y) and L(X, Y ). In a * similar way, the operator RB : Y ! X which satises hx, RB(y)i = B(x, y) provides an isometric isomorphism * between B(X × Y) and L(Y, X ) via B $ RB. Hence we have * ∼ ∼ * ∼ * (X⊗ˆ π Y) = B(X × Y) = L(X, Y ) = L(Y, X ). (1) An element u 2 X ⊗ Y can be considered as a bilinear form Bu on X* × Y* by m X Bu(φ, ψ) = φ(xi)ψ(yi), i=1 u Pm x ⊗ y u X ⊗ Y B X* where = i=1 i i is any representation of . This provides a canonical embedding of into ( × Y*), and we use it to dene what is called the injective norm on X ⊗ Y, which is the norm induced by this embedding.