The dual of the space of bounded operators on a Banach space


 Given Banach spaces X and Y, we ask about the dual space of the 𝒧(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
 *.


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 = ∞ ( ), where X is any Banach space. Clearly T ∈ L(X, ∞ ( )) can be written as Tx = (φ (x), φ (x)) for x ∈ X and where φ , φ are elements of X * . Furthermore, T = max{ φ , φ }. It follows that if ψ ∈ L(X, Y) * , then ψ = (ψ , ψ ) where ψ , ψ ∈ X ** and ψ = ψ + ψ . Thus, the dual space in this case is the space ( , 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, L A (X, Y) for the approximable operators (those for which the nite rank operators are dense), and L K (X, Y) for the compact operators from X to Y. The closed unit ball of a Banach space X is denoted by B X .
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)) * = (n, X ** ) which we described above could be expressed by L(X, ∞ (n)) * = X **⊗ π ( ∞ (n)) * . Before showing how that works, let us take a little refresher course on tensor products.

Tensor products
We mention here some basic and useful facts about tensor products for the bene t 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 ∈ X and y ∈ Y, let x ⊗ y denote the linear functional on B(X × Y) de ned by 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 u = n i= λ i x i ⊗ y i and since such representations are not unique, we may always write an element in the form The projective norm, π, on the tensor product of two Banach spaces X ⊗ Y is de ned by The in mum is taken, of course, over all representations of u. Then π is a norm with the property that π(x⊗y) = x y for x ∈ X and y ∈ Y. The tensor product X ⊗ Y with the projective norm is denoted by X ⊗π Y and its completion is given by X⊗ π Y. Some useful facts for future reference are that ⊗ π Y = (Y) and this holds also for the space (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 o er a proof, although we give an alternative to the interesting one given in [7]. Theorem 1. Let X and Y be Banach spaces, u ∈ X⊗ π Y, and ϵ > . Then there exist bounded sequences {xn}, {yn} in X, Y respectively, such that the series ∞ n= xn ⊗ yn converges to u and ∞ n= xn yn < π(u) + ϵ.
It follows from the above result that where the in mum 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 > such that |B(x, y)| ≤ C x y for all x ∈ X and y ∈ Y. Let B(X, Y) denote the space of bounded bilinear forms on X × Y, which is a Banach space with the norm Given B ∈ B(X × Y) there exists a unique linear functionalB on X⊗ π Y satisfyingB(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⊗ π Y) * . We note also, that given B ∈ B(X × Y), we can de ne an operator L B : X → Y * by y, L B (x) = B(x, y) and the correspondence B ↔ L B describes an isometric isomorphism between B(X × Y) and L(X, Y * ). In a similar way, the operator R B : Y → X * which satis es x, R B (y) = B(x, y) provides an isometric isomorphism between B(X × Y) and L(Y , An element u ∈ X ⊗ Y can be considered as a bilinear form Bu on X * × Y * by where u = m i= x i ⊗ y i is any representation of u. This provides a canonical embedding of X ⊗ Y into B(X * × Y * ), and we use it to de ne what is called the injective norm on X ⊗ Y, which is the norm induced by this embedding. We denote by ε(u) the injective norm of u ∈ X ⊗ Y. Hence, we have where the supremum is taken over all representations of u. By consideration of the operators Lu : X * → Y and Ru : Y * → X, we have alternate forms for ε(u) given by We will denote by X ⊗ε Y the tensor product X ⊗ Y with the injective norm and its completion (which is called the injective tensor product) by X⊗ ε Y. This injective tensor product may be considered as a subspace of B(X * × Y * ) or of L(X * , Y) or L(Y * , X). Of particular interest to us is that It is the case that ε(u) ≤ π(u) for every u ∈ X ⊗ Y and ε(x ⊗ y) = x y for every x ∈ X and y ∈ Y. It can be shown that c ⊗ ε X = c (X) and ⊗ ε X = [X], where [X] is the space of all unconditionally summable sequences in X. Furthermore, the injective tensor product of C(K) with a Banach space X, where K is a compact Hausdor space, can be identi ed with the space C(K, X) of continuous functions from K into X.
If we consider u in X ⊗ε Y as a continuous function on the product space B X * × B Y * , which is compact when endowed with the weak * topology on each unit ball, then we can think of X⊗ ε Y as embedded into Since we know the description of such functionals, we have the following theorem, which describes the dual space of X⊗ ε Y.
for every x ∈ X, y ∈ Y. Furthermore, the norm ofB is given by where µ ranges over the set of all measures that correspond to B in this way, and this in mum is attained.
A bilinear form B on X × Y is called an integral bilinear form ifB, as in the theorem, is a bounded linear functional on X⊗ ε Y, with integral norm de ned by where the in mum is taken over all the regular Borel measures on B X * × B Y * that satisfy (3). The Banach space of integral bilinear forms with integral norm is denoted by B I (X × Y). Hence we have We mention here that a bounded bilinear form B on X × Y is said to be nuclear if and only if there exist bounded sequences {φn} in X * and {ψn} in Y * which satisfy ∞ n= φn ψn < ∞, such that for every x ∈ X, y ∈ Y. An expression ∞ n= φn ⊗ ψn is called a nuclear representation of B. The nuclear norm of B is de ned by It is worth noting that every nuclear bilinear form is also integral. If B(x, y) = ∞ n= λn φn(x)ψn(y), where {λn} is a summable sequence of scalars, and {φn}, {ψn} are sequences of distinct unit vectors in X * , Y * respectively, the sum may be interpreted as an integral like that in (3), where µ is the measure with point masses λn at the points (φn , ψn).
The notions of integral and nuclear can also be applied to operators. An operator T : X → Y corresponds to a bilinear form B T de ned on X × Y * by B T (x, ψ) = Tx, ψ . Then T is said to be integral if B T is integral, and the integral norm T I is that of B T . The space of integral operators with that norm is denoted by I(X, Y). An operator from X to Y is said to be nuclear if it is in the range of the canonical operator N : X *⊗ π Y → L(X, Y) of unit norm that associates the tensor u = ∞ n= φn ⊗ yn with the operator Lu given by Lu(x) = ∞ n= φn(x)yn.
The collection of such operators is written as N(X, Y), and the nuclear norm of such an operator T is de ned as where the in mum is taken over all such representations of T where {φn}, {yn} are sequences in X * , Y respectively such that ∞ n= φn yn < ∞. We note for future reference that the map N is not necessarily one-to-one.
From (1) we remember that B(X × Y) is isometrically isomorphic to the space L(X, Y * ), where a bilinear form B is associated with the operator T by B(x, y) = y, Tx . Such a form B is integral if and only if the associated T is integral and the two share their integral norms. This pairing establishes an isometric isomorphism of B I (X × Y) with I(X, Y * ). This allows us to say that where we use "equal" signs rather than the more precise congruences.

Dual spaces
In thinking about the dual space of L(X, Y), we are drawn to the fact that the injective tensor product X *⊗ ε Y can be regarded as a subspace of L(X, Y) as in (2). In fact, we can write X *⊗ ε Y = L A (X, Y), the space of approximable operators. Our main theorem will be a theorem recorded by Ryan [7,Theorem 5.33] that gives a special characterization of the dual of X⊗ ε Y under certain conditions on X and Y. Our proof, though put together a bit di erently, is certainly in uenced by that of Ryan. We recall that a Banach space X is said to have the approximation property if for every Banach space Y, every operator T : X → Y may be approximated on compact sets by nite rank operators. Equivalently, X has the approximation property if and only if for every Banach space Y, every compact operator from Y into X is approximable. In addition, we need to use the notion of a Banach space having the Radon-Nikodým property. A vector measure µ on a σ-algebra Σ with values in a Banach space X satis es the Radon-Nikodým Theorem if µ has bounded variation and is absolutely continuous with respect to a nite positive measure λ, then µ is an inde nite Bochner integral with respect to λ. We say X has the Radon-Nikodým property if every such measure as above with values in X has the Radon-Nikodým property.
Before stating and proving the theorem just mentioned, we need two lemmas, also given by Ryan [7].

Lemma 3. Let Σ be a σ-algebra of subsets of Ω, M(Σ) the Banach space of scalar measures on Σ with the variation norm, and X a Banach space. Then M(Σ)⊗ π X is isometrically isomorphic to the space M (Σ, X) of vector measures with the Radon-Nikodým property, with the variation norm µ = |µ| (Ω).
Proof.
It is known that π(u) = µ and that the correspondence is an isometry. It remains to show that M(Σ) ⊗ X is dense in M (Σ, X). Let µ ∈ M (Σ, X). There exists f ∈ L (|µ| , X) such that µ(E) = E fd|µ| for all E ∈ Σ. Since L (|µ| , X) = L (|µ| )⊗ π X, there exist bounded sequences {gn}, {xn} in L (|µ| ), X respectively such that ∞ n= gn xn < ∞ and f = ∞ n= gn ⊗ xn. Let µn be de ned by µn(E) = E gn d|µ| for each n. Then µ corresponds to ∞ n= µn ⊗ xn ∈ M (Σ, X). Before stating the next lemma, we want to de ne what is called a representing measure. If T is a bounded operator on the space C(K) of continuous functions on a compact, Hausdorf space K to X, we may de ne a function µ T on the σ-algebra of Borel subsets of K with values in the bidual X ** by µ T (E) = T ** (χ(E)). Then for each φ ∈ X * we have where T * φ ∈ C(K) * is a regular Borel measure on K. The measure µ T is called the representing measure for T. On the other hand, given a regular vector measure µ on the Borel sets of K with values in X ** , we have an associated operator Tµ de ned on X * by Tµ φ = φµ. The scalar measure φµ is regular whenever µ is regular.

Lemma 4. Let T : C(K) → X be an operator with representing measure µ. Then T is a nuclear operator if and only if µ has the Radon-Nikodým property. If T is nuclear, then T N = µ .
Proof. It is known that if T is an operator on a space whose dual space has the approximation property, then T * is nuclear if and only if T is nuclear. Since T : C(K) → X in this case, we know that C(K) * is the space of regular Borel measures on the σ-algebra B K of Borel sets in K, and since this space has the approximation property, we have T is nuclear if and only if T * is nuclear. Furthermore, it is easily shown that C(K) * is complemented in M(B K ) by a projection of norm one. If I denotes the embedding of C(K) * into M(B K ), then we see that IT * is the operator associated with µ, so that IT * (φ) = φµ for all φ ∈ X * . Supppose µ has the Radon-Nikodým property. We have from Lemma 3 that M(B K )⊗ π X = M (B K , X). Therefore, µ can be thought of as a member of the projective tensor product, so we can write where ∞ n= µn xn < ∞. If we write S = IT * , then S(φ) = φµ and where J is the canonical mapping from X into X ** . Let u ∈ X **⊗ π M(B K ) be given by u = ∞ n= Jxn ⊗ µn. We see that the canonical map N from X **⊗ π M(B K ) to L(X * , M(B K )) de ned by Nu(φ) = ∞ n= Jxn(φ)µn agrees with S so that S (and therefore T * ) is nuclear.
On the other hand, suppose T and therefore T * is nuclear. Then S = IT * is nuclear and must be de ned as above. It follows that µ must be in M (B K ), X ** ) and so satis es the Radon-Nikodým property.
At this point we are ready to state and prove the theorem we mentioned at the beginning of this section. Theorem 5. Let X and Y be Banach spaces for which X * has the Radon-Nikodým property and either X * or Y * has the approximation property. Then Proof. In this proof, we will assume that Y * has the approximation property. We begin by considering the canonical map N from X *⊗ π Y * to L(Y , X * ) which we have considered in the discussion of nuclear operators. If we can show that N is one-to-one, then we will establish that the space N(Y , X * ) is isometrically isomorphic to X *⊗ π Y * . Let u = ∞ n= φn ⊗ ψn ∈ X *⊗ π Y * , where φn ∈ X * , ψn ∈ Y * , and ∞ n= φn ψn < ∞. Then Nu(y) = ∞ n= ψn(y)φn. Suppose Nu = . We may assume that ψn → and ∞ n= φn < ∞, for if not, we can alter the representation of u so that both hold. Let T ∈ L(Y * , X ** ) and ϵ > be given. Since {ψn} is a zero convergent sequence in Y * , it is relatively compact and because Y * has the approximation property, there is a nite rank operator S such T(ψn) − S(ψn) < ϵ for every n. The operator S has the form S(y * ) = m i= ψ * i (y * )φ * i , where ψ * i ∈ Y ** , φ * i ∈ X ** for i = , . . . , m. We have S ∈ (X *⊗ π Y * ) * = L(Y * , X ** ), so that The zero occurs in the last line above since we know that ∞ n= ψ(y)φn = for every y ∈ Y. This is enough to show that ∞ n= y ** (ψ i )φn = for every y ** ∈ Y ** since {Jy : y ∈ Y} is w * -dense in Y ** . We conclude that u, S = . This allows us to write Since ∞ n= φn < ∞ and ϵ is arbitrary, we must have u, T = for all T from which we conclude that u = and N is injective. We remark that if the assumption is that X * has the approximation property, in the above argument let T ∈ L(X * , Y ** ) and proceed in the same way.
From (4), we see that the proof will be complete if we can show that any T ∈ I(Y , X * ) is also in N(Y , X * ). Given such a T, we know it corresponds to a bilinear form B ∈ B I (X ⊗ Y) by x, Ty = B(x, y). Because B is integral, there is a regular Borel measure ν on K = B x * × B Y * such that By the Radon-Nikodým Theorem there is a Borel measurable function h such that |h(t)| = for all t ∈ K such that dν = hd|ν|, where |ν| is the total variation of ν. Let µ = |ν|, S : Y → C(K), de ned by Sy(φ, ψ) = ψ(y) and R : X → L ∞ (µ) given by Rx(φ, ψ) = φ(x). This gives Let V denote the restriction of R * to L (µ) (really JL (µ)), and let I denote the canonical embedding of C(K) into L (µ). By the duality between L (µ) and L ∞ (µ), we can see that Rx, ISy = x, VISy = x, Ty .
Thus ψn ≤ µn , so ∞ n= ψn φn < ∞. Let w = ∞ n= ψn ⊗ φn ∈ Y *⊗ π X * . We have Hence, Nw = T and T ∈ N(Y , X * ). We have noted before that nuclear bilinear forms are integral, and the same holds for nuclear operators. This completes the proof.

Corollary 6.
If either X * or Y * has the Radon-Nikodým property and either X * or Y * has the approximation property, then If both X * and Y * have the Radon-Nikodým property and either X * or Y * has the approximation property, then X *⊗ π Y * has the Radon-Nikodým property. (See [2, p.29].) We now turn to the original question. What can we say about the dual of L(X, Y)? We begin with the nite dimensional case. We recall from (2) that X *⊗ ε Y is contained in L(X, Y) and equal to L A (X, Y). Since this space is equal to L(X, Y) if either X or Y is nite dimensional, we have the following results.

Theorem 7. Suppose X and Y are Banach spaces. (i) If Y is nite dimensional, then
Proof. Since nite dimensional spaces possess both the Radon-Nikodým property and the approximation property, the results follow immediately from Corollary 6.
Proof. Since both spaces involved satisfy both the Radon-Nikodým property and the approximation property, and since every operator from q to p is compact (by Pitt's Theorem), the result follows from Corollary 6.
Since every operator from c to is compact and has the Radon-Nikodým property, we also have We note here that if either X * or Y has the approximation property, then X *⊗ ε Y = L K (X, Y), the compact operators from X to Y. Hence the following theorem follows from Corollary 6.
Theorem 10. (Grothendieck [3]. (See also [6].) If either X ** or Y * has the Radon-Nikodým property and either has the approximation property, then The three theorems previous to Theorem 10 were, of course, actually included in this last one, since in each such case, the bounded operators from X to Y were all compact, hence as we said in the beginning, "probably in Grothendieck." The most general statement we can make concerning duals of bounded operators is that L(X, Y * ) * = (X⊗ π Y) ** .
This is most useful when we can describe the projective tensor product of X and Y. Since we know that ⊗ π X = (X) and ⊗ π = , we can state the following.

More on Tensor products
It was shown in Ryan's book, [7, p. 23], that the tensor diagonal subspace of ⊗ π is isometrically isomorphic to . Actually, it is a 1-complemented subspace. The following is also true.
Theorem 13. Let < p, q < ∞ with p and q conjugate. Then is isometrically embedded in p⊗ π q .
Proof. The argument follows as in Ryan's book. We denote by {en} and {fn} the standard Schauder bases for p and q respectively. We consider the subspace W spanned by en ⊗ fn. An arbitrary element u in W is of the form u = ∞ i= αn en ⊗ fn. First, we have π(u) ≤ ∞ n= |αn|. Towards the reversed inequality we de ne B(x, y) = ∞ n= sgn(αn)xn yn which implies that |B(x, y)| ≤ x p y q , from an application of the Hölder inequality. We recall that sgn(αn) is a modulus 1 complex number such that sgn(αn)αn = |αn| if αn ≠ , otherwise, it is equal to zero. Since B ≤ we have π(u) ≥ u, B = ∞ n= |αn|. This implies that π(u) = ∞ n= |αn|. Therefore W is isometrically isomorphic to , implying that is isometrically embedded in p⊗ π q . As in the case, is a 1-complemented subspace in p⊗ π q .
The only linear mapping determined by the bilinear map P(x, y) = ∞ n= xn yn en ⊗ fn is a norm 1 projection. Several observations can be derived from this fact: