Trace-Class and Nuclear Operators

This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.


Introduction
This is an expository paper on trace-class and nuclear operators. Its purpose is to demonstrate that these classes of operators coincide on a Hilbert space. It will focus mainly on three points: (i) where these notions came from, (ii) how they are intertwined, and (iii) when they coincide. Being a plain expository paper, this has no intention to survey the subject, neither to offer an extensive bibliography on it.
To begin with, let us borrow the definitions of nuclear and trace-class operators from Sections 4 and 5 (where these definitions will be properly posed).
• Nuclear Operators. An operator T on a Banach space X is nuclear if there are X * -valued and X -valued sequences {f k } and {y k } such that k f k y k < ∞ and T x = k f k (x)y k for every x ∈ X . • Trace-Class Operators. An operator T on a Hilbert space X is traceclass if γ |T |e γ ; e γ < ∞ for an arbitrary orthonormal basis {e γ } for X and the series value does not depend on {e γ }.
Perhaps the first lines in Robert Bartle's review of [9] might be a suitable start: "Grothendieck [8] showed that a Banach space X has the approximation property if and only if, for every nuclear operator T : X → X (i.e., operator having the form T = k f k ; · y k ) with f k ∈ X * , y k ∈ X , and k f k y n < ∞), the number tr(T ) = k f k , y k is well-defined (i.e., is independent of the choice of {f k } and {y k } in the representation T = k f k ; · y k ) and can be used to define a trace function." Bartle's concise description nicely summarizes the apparently long way to be covered from Grothendieck's projective tensor products (where nuclear operators originate form), to trace-class operators, and finally concluding that these classes coincide. The familiar notion of trace as the sum of eigenvalues is a fundamental result known as Lidskiǐ Theorem [17], [7,Theorem 8.4] which still remains an active research topic (e.g., [23,11,6,3,25,20]).
These concepts (nuclear and trace-class) are linked together since their early days. Schatten in his celebrated 1950 monograph [27] (which actually is an offspring of his 1942 thesis) describes nuclear Hilbert-space operators as being precisely the trace-class: "The trace-class may be also interpreted as X ⊗ ∧ X * " [27, Theorem 5.12] -the completion of the tensor product of a Hilbert with its dual, with respect to the greatest reasonable crossnorm (i.e., the projective norm).
The paper is organized as follows. Section 2 summarizes some common notation and terminology. Section 3 poses the necessary results on crossnormed tensor products of a pair of Banach spaces, since this is the proper setup where nuclear transformations come from. Nuclear transformations are defined in Section 4 (Theorem 4.1) as the range of a linear contraction of the projective tensor product X * ⊗ ∧ Y into the Banach space B[X , Y] of all bounded linear transformations from X to Y, which yields the characterization of nuclear operators mentioned above. Section 5 deals exclusively with trace-class operators on a Hilbert space, and gives a thorough view of basic properties of these operators. Section 6 shows in Theorem 6.1 that nuclear and trace-class operators in fact reduce to the same thing.
All terms and notation above will be defined here in due course.

Notation and Terminology
Throughout the paper all linear spaces are over the same field F (and the field F in this context means either R or C). If X , Y, Z are linear spaces, then let L[X , Z] and b[X ×Y, Z] denote the linear spaces of all linear transformation of X into Z, and of all bilinear maps of the Cartesian product X ×Y into Z. If X , Y, Z are normed spaces, then let B[X , Z] denote the normed space of all bounded (i.e., continuous) linear transformations of X into Z equipped with its standard induced uniform norm, and let b[X ×Y, Z] denote the normed spaces of all bounded (i.e., continuous) bilinear maps of X ×Y into Z equipped with its usual norm, which are both Banach spaces whenever Z is (see e.g., [16]).
A subspace of a normed space is a closed linear manifold of it. If X is a normed space, then X * = B[X , F] stand for its dual; if M is a subset of an inner product space, then M ⊥ stands for the orthogonal complement of M. Range and kernel of a bounded linear transformation T ∈ B[X , Y] between normed spaces X and Y will be denoted by R(T ) -a linear manifold of Y -and N (T ) -a subspace of X -respectively. If two normed spaces X and Y are isometrically isomorphic, and if y ∈ Y is the isometrically isomorphic image of x ∈ X , then write X ∼ = Y and x ∼ = y.
Let X and Y be arbitrary normed spaces. By an operator we mean a bounded linear transformation of a normed space into itself. Set B[X ] = B[X , X ]; the normed algebra of all operators on X . Let B 0 [X , Y] and B ∞ [X , Y] stand for the normed spaces of all bounded finite-rank (i.e., dim R(T ) < ∞) and of all compact linear transformations T : ; the ideals of the algebra B[X ] consisting of all bounded finiterank and of all compact operators, respectively, so that ) stand for the normed spaces of all nuclear transformations. It is worth noticing that there are different, also common, notations such as K for compact, F for finite-rank, and N for nuclear operators -at the end of Section 5 it will become clear the reason for our choice in favor of the above sub-indexed-B notation (see also, e.g., [29, Sections 6.1 for every normed space X -every Banach space with a Schauder basis has the approximation property, in particular, since the range of a compact linear transformation is separable, every Hilbert space has the approximation property.

Preliminaries on Crossnormed Tensor Product Spaces
The algebraic tensor product of linear spaces X and Y is a linear space X ⊗ Y for which there is a bilinear map θ : X ×Y → X ⊗ Y (called the natural bilinear map associated with X ⊗ Y) whose range spans X ⊗ Y with the following additional property: for every bilinear map φ : X ×Y → Z into any linear space Z there exists a (unique) linear transformation Φ : X ⊗ Y → Z for which the diagram commutes. Set x ⊗ y = θ(x, y) for each (x, y) ∈ X ×Y. These are the single tensors.
An arbitrary element ̥ in the linear space X ⊗ Y is a finite sum i x i ⊗ y i of single tensors, and the representation of ̥ as a finite sum of single tensors ̥ = i x i ⊗ y i is not unique. (For an exposition on algebraic tensor products see, e.g., [15].) If X and Y are Banach spaces and X * and Y * are their duals, then let x ⊗ y and f ⊗ g be single tensors in the tensor product spaces X ⊗ Y and It can be verified that (i) x ⊗ y α = x y whenever · α is a reasonable crossnorm, and (ii) when restricted to X * ⊗ Y * the norm · * α on (X ⊗ Y) * is a reasonable crossnorm (with respect to (X * ⊗ Y * ) * ). Two special norms on X ⊗ Y are the so-called injective · ∨ and projective · ∧ norms, where the infimum is taken over all (finite) representations of ̥ ∈ X ⊗ Y. It can also be shown that (iii) these are indeed norms on X ⊗ Y, that (iv) both · ∨ and · ∧ are reasonable crossnorms and, moreover, that (v) a norm · α on X ⊗ Y is a reasonable crossnorm if and only if Let X ⊗ α Y = (X ⊗ Y, · α ) stand for the tensor product space of a pair of Banach spaces equipped with a reasonable crossnorm · α , which is not necessarily complete. Their completion (see, e.g., [13,Section 4.7]) is denoted by X ⊗ α Y (same notation · α for the extended norm on X ⊗ α Y). In particular, X ⊗ ∨ Y and X ⊗ ∧ Y are referred to as the injective and projective tensor products. For the theory of the Banach space X ⊗ α Y (in particular, X ⊗ ∨ Y and X ⊗ ∧ Y) see, e.g., [10], [2], [26], [4]. The next two fundamental results on the projective tensor product will be need along the next two sections. The first one is referred to as Grothendieck Theorem (see, e.g., [ Theorem 3.1. (Grothendieck). If X and Y are Banach spaces, then for every ̥ ∈ X ⊗ ∧ Y there exist X -valued and Y-valued sequences {x k } and {y k }, respectively, for which the real sequence { x k y k } is summable and ̥ = k x k ⊗ y k (i.e., ̥ ∈ X ⊗ ∧ Y is representable in the form of a countable sum ̥ = k x k ⊗ y k in the sense that it is either a finite or countably infinite representation). Moreover, the projective norm · ∧ on X ⊗ ∧ Y is given by where the infimum is taken over all representations k Identify X ⊗ ∧ Y with isometrically isomorphic images of it, and so regard X ⊗ ∧ Y as being dense in X ⊗ ∧ Y. Thus take ̥ ∈ X ⊗ ∧ Y \X ⊗ ∧ Y (otherwise the resulting finite sum is trivially obtained) so that ̥ is arbitrarily close to elements in X ⊗ ∧ Y. Take an arbitrary ε > 0. For each posi- Thus the sequence {x i ⊗y i } is absolutely summable with i x i ⊗ y i ∧ < ̥ ∧ + 2ε. Therefore, since X ⊗ ∧ Y is a Banach space, the sequence is summable. This means the sequence of partial sums { Proof. Take the natural bilinear map θ ∈ b[X ×Y, X ⊗ Y] associated with the tensor product space X ⊗ Y. It is readily verified that the composition with θ, namely , which remains linear and injective, Since this holds for every (finite) representation of and the linear Φ is bounded: Thus, as this holds for every (b) Take an arbitrary Φ ∈ B[X ⊗ ∧ Y, Z]. As we saw in ( * ), by (a 1 ) and so, as we saw in ( * * ), So the stated result follows by transitivity.

Nuclear Operators
Let X , Y, V be Banach spaces. As a starting point, consider the following expression involving the projective tensor product.
The above inclusion comes from the definition of a reasonable crossnorm, the first isometric isomorphism is a particular case of the universal mapping principle for the projective norm as in Theorem 3.2, and the second one is the classical identification of bounded bilinear forms with bounded linear transformations (see, e.g., [2, Section 1.4, p.6]). For the particular case where Y is isometrically isomorphic to the dual of some Banach space where the inclusion means algebraic embedding. On the other hand, this not only holds in general (no restriction to Banach spaces being the dual of some Banach space) but is strengthened to an isometric embedding for the injective norm instead: the injective tensor product X * ⊗ ∨ Y is isometrically embedded in the Banach space B[X , Y], and so it is viewed as a subspace of B[X , Y]: (see, e.g., [4, Proposition 1.1.5]). This fails in general for the projective norm, and the theorem below shows how far one can get along this line in the general case.
if and only if there are X * -valued and Y-valued sequences {f k } and {y k } for which the real sequence { f k y k } is summable and and (c) for each T ∈ R(K), Proof. Let X * be the dual of X . Take an arbitrary which is linear and does not depend on the representation for every x ∈ X , since K(̥) = k K(f k ⊗ y k ) because K is linear and bounded. This characterizes the range R(K) of K.
(c) As for · N being a norm on R(K), we verify the triangle inequality only.
Theorem 4.1 prompts the following redefinition: The expression T x = k f k (x)y k is a nuclear representation of T and (where the infimum is taken over all nuclear representations of T ) defines a norm on the linear space B N [X , Y], the nuclear norm, such that T ≤ T N .
The linear contraction K of norm one in Theorem 4.1 is not necessarily injective (thus it may not be an isometry) and R(K) is identified with the quotient space of Moreover, B N [X ] is a two-sided ideal of the Banach algebra B[X ] by the next result.
Proof. This is a straightforward consequence of the definition of nuclear transformation: as k f k y k < ∞ and T From now on let X be a Hilbert space. Denote the inner product in X by · ; · , and take the Banach algebra B[X ] of all operators on X . Consider the Fourier Series and the Riesz Representation Theorems (see, e.g., [13,Theorems 5.48 and 5.62]). A functional f lies in X * if and only if there is a unique z in X , called the Riesz representation of f , such that f (x) = x ; z for every x ∈ X and z = f . Thus on a Hilbert space the previous redefinition can be rewritten as follows: An operator T ∈ B[X ] on a Hilbert space X is nuclear (i.e., T ∈ B N [X ]) if there are X -valued sequences {z k } and {y k } such that k z k y k < ∞ and T x = k x ; z k y k for every x ∈ X .
For each T ∈ B[X ] let T * ∈ B[X ] stand for its Hilbert-space adjoint. T * x = k x ; y k z k for every x ∈ X .
(b) T is nuclear if an only if its adjoint T * is nuclear and T * N = T N .
Proof. Let T ∈ B[X ] be an operator on a Hilbert space X .
(a) Suppose there are X -valued sequences {z k } and {y k } such that Then T x ; y = k x ; z k y k ; y = k x ; z k y k ; y = k x ; y ; y k z k = x ; k y ; y k z k for every x, y ∈ X , and therefore (by uniqueness of the adjoint) T * y = k y ; y k z k for every y ∈ X .
(b) Immediate from item (a) and the definition of nuclear operator on X .

Trace-Class Operators
Let X be a Hilbert space. A summary of the elementary expressions required in this section goes as follows. For each T ∈ B[X ] set |T | = (T * T ) x . Now let {e γ } γ∈Γ and {f γ } γ∈Γ be arbitrary orthonormal bases for a Hilbert space X , indexed by an arbitrary nonempty index set Γ (alternate notation: {e γ } γ or {e γ }). By the Parseval identity (viz., whenever any of the families { T e γ } γ or { T * f γ } γ is square summable (i.e., if γ T e γ 2 < ∞ or γ T * f γ 2 < ∞) for some orthonormal bases for X . Applying the above displayed identity to |T | Thus if the family of positive numbers |T |e γ ; e γ γ = |T | 1 2 e γ 2 γ is summable, then its sum does not depend on the orthonormal basis {e γ } γ for X .
An operator T ∈ B[X ] on a Hilbert space X is trace-class if γ |T |e γ ; e γ < ∞ for an arbitrary orthonormal basis {e γ } for X .
and so (recall: |T | = |T |) we may infer: Lemma 5.1. Let X be a Hilbert space. The following assertions hold true.
Since homogeneity, nonnegativity and positivity for · 2 : B 2 [X ] → R is readily verified, Claim 1 is enough to ensure that is a linear space and · 2 is a norm on it.
is an ideal (i.e., a two-sided ideal) of B[X ].
Now take an arbitrary T ∈ B 2 [X ] so that γ∈Γ T * e γ 2 = γ∈Γ T e γ 2 < ∞, and take any integer n ≥ 1. Thus there is a finite set N n ⊆ Γ such that k∈N T * e k 2 < 1 n for all finite sets N ⊆ Γ\N n (Cauchy criterion for summable families -see, e.g., [13,Theorem 5.27]). So γ∈Γ\Nn T * e γ 2 < 1 n . Recall that T x = γ∈Γ T x ; e γ e γ for every x ∈ X (Fourier series expansion). Set T n x = k∈Nn T x ; e k e k for each x ∈ X , which defines an operator T n in B 0 [X ] because N n is finite, Hence (T − T n )x 2 = γ∈Γ\Nn | T x ; e γ | 2 ≤ γ∈Γ\Nn T * e γ 2 x 2 for every x ∈ X . This implies that T n − T → 0, and so T is the uniform limit of a sequence of finite-rank operators on a Banach space, and therefore T is compact (see, e.g., [13,Corollary 4.55]).
Every Hilbert-Schmidt operator is compact.
As we saw above, γ T e γ 2 = γ T * e γ 2 and T e ≤ T 2 if e = 1, and so The norm · 2 on the linear space B 2 [X ] of all Hilbert-Schmidt operators is referred to as the Hilbert-Schmidt norm.
Theorem 5.2. Let X be a Hilbert space. The following assertions hold true.
(a) The set B 1 [X ] is a linear space and · 1 :    Proof of Claim 2. Consider the polar decompositions T + S = W |T + S|, T = W 1 |T | and S = W 2 |S|, where W,W 1 ,W 2 are partial isometries in B[X ], so that |T + S| = W * (T + S), |T | = W * 1 T , and |S| = W * 2 T . Hence (Schwartz inequality on ℓ 2 over Γ)  (by the Schwarz inequality on both Hilbert spaces X and ℓ 2 over Γ, as we did before).
Also, since the product AB lies in B 2 [X ] by Lemma 5.1(c), Every trace-class operator is Hilbert-Schmidt.
Moreover, since |T |x 2 ≤ T |T | 1 2 x 2 for every x ∈ X and T ≤ T 2 (by Lem- 2 W * 1 ≤ T 1 (by the Schwarz inequality on both Hilbert spaces X and ℓ 2 over Γ again). Thus { T e γ ; e γ } is a summable family, and by taking an arbitrary orthonormal basis {f γ } for the Hilbert space X , and applying the Fourier Series Theorem, we get By (i 1 ) and (i 2 ) we get (i). As S T and T S lie in B 1 [X ] by (d), it follows by (i) that the sums in (ii) exist and do not depend on the orthonormal basis. So let {f γ } be any orthonormal basis for X and consider again the Fourier Series Theorem. Thus Let {e δ } be an orthonormal basis for N (T * ) and let {e k } be a finite orthonormal basis for N (T * ) ⊥ . As X = N (T * ) + N (T * ) ⊥ , then {e γ } = {e δ } ∪ {e k } is an orthonormal basis for X . Now either T * e γ = 0 or T * e γ = T * e k . Therefore γ |T * |e γ ; e γ = k |T * |e k ; e k < ∞. Thus Every finite-rank operator is trace-class.
To proceed we need the following auxiliary result which will support Remark 5.4 and Theorem 6.1. It is a standard application of the Spectral Theorem for compact operators (for similar versions see, e.g., [19, Theorem 6.14.1], [29,Theorem 7.6]). Proposition 5.3. If T is compact, then there exist an orthonormal basis {e γ } for X and a family of nonnegative numbers {µ γ } such that |T |x = γ µ γ x ; e γ e γ for every x ∈ X .
Proof. The operator |T | ∈ B[X ] is nonnegative (so normal) and compact. (As the class of compact operators from B[X ] is an ideal of B[X ], the nonnegative square root |T | of the nonnegative compact |T | 2 is compact since |T | 2 = T * T is compact -see e.g., [13,Problem 5.62].) Since |T |x = T x for every x ∈ X we get N |T | = N (T ). Then by the Spectral Theorem there is a countable orthonormal basis {e k } for the separable Hilbert space H = N (T ) ⊥ consisting of eigenvectors of |T | associated with positive eigenvalues {µ k } of |T | such that |T |u = k µ k u ; e k e k for every u ∈ H (see, e.g., [14,Corollary 3.4]). Since X = H ⊕ N with N = N (T ), there is an orthonormal basis {e γ } = {e k } ∪ {e δ } for X . Here {e δ } is an orthonormal basis for the (not necessarily separable) Hilbert space N , where |T |v = 0 for v ∈ N so that the above expansion on H describes |T |x for all x = u ⊕ v in the orthogonal direct sum X = H ⊕ N (with |T |e δ = 0, T e k = µ k e k , and µ γ = 0 if γ = k).
Again, suppose γ |T |e γ ; e γ < ∞, which means T lies in B 1 [X ]. By Theorem 5.2(b) T = AB with A, B ∈ B 2 [X ] and so γ Ae γ 2 < ∞ and γ Be γ 2 < ∞ for an arbitrary orthonormal basis {e γ } for X . Since 2| T e γ , e γ | = 2| ABe γ ; e γ | = 2| Be γ ; A * e γ | ≤ 2 Be γ A * e γ ≤ Be γ 2 + A * e γ 2 , we get 2 γ | T e γ , e γ | ≤ (Actually, Claim 3(i) in the proof of Theorem 5.2 has shown by a different proof that T ∈ B 1 [X ] implies γ | T e k ; e γ | < T 1 ). However, the converse of (b) fails: (c) γ | T e k ; e γ | < ∞ for every orthonormal basis {e γ } =⇒ T ∈ B 1 [X ]. Indeed, take a unilateral shift S ∈ B[X ] of multiplicity one on an infinite-dimensional separable Hilbert space X . Then S shifts a countable orthonormal basis for X . Say Se k = e k+1 for each integer k > 0 for some orthonormal basis {e k } for X . Observe that Sf k ; f k = 0 for every orthonormal basis {f k } for X . In fact, take any orthonormal basis {f k } for X and consider the Fourier expansion of f k in terms of {e k }, viz., f k = j f k ; e j e j , and so Sf k = j f k ; e j Se j . Then e j+1 ; f k e j ; f k = e j+1 ; e j = 0, by taking the Fourier expansion of each e k in terms of {f k }. Thus k | Sf k ; f k | = 0 for every orthonormal basis {f k }. But S is an isometry, so S * S = I and hence |S| = I, the identity on X . Thus S ∈ B 1 [X ] (it is not even compact).
Theorem 5.5. Let X be a Hilbert space. The following assertions hold true.
Proof. (a) Essentially the same argument that proves completeness of (ℓ 1 , · 1 ). Let {T n } be an arbitrary B 1 [X ]-valued Cauchy sequence in (B 1 [X ], · 1 ). Then it is a Cauchy sequence in the Banach space (B[X ], · ) (since · ≤ · 1 ), and so Recall that the product of a pair of uniformly convergent sequences of operators converges uniformly to the product of the limits, and also that uniform convergence is preserved both under the adjoint and under the square root operations (see, e.g., [13,Problems 4.46,5.26,5.63]). Thus T n − T = |T n − T | Let {e γ } γ∈Γ be any orthonormal basis for X and take an arbitrary T ∈ B 1 [X ]. So where the supremum is taken over all finite sets J ⊆ Γ. Take an arbitrary ε > 0. The above expression (asserting that the family { T e γ , e γ } γ is summable) ensures the existence of a finite set J ε ⊆ Γ for which γ∈Γ\Jε |T |e γ ; e γ < ε. Then set X ε = span {e γ ∈ X : γ ∈ J ε }, a finite-dimensional subspace of X , and take |T |e γ ; e γ < ε, Either Claim 3 in the proof of Theorem 5.2 or Remark 5.4(b) ensure that the family { T e γ , e γ } γ is summable (i.e., the series { γ T e γ , e γ } γ converges in (F, | · |)) for every orthonormal basis {e γ } for the Hilbert space X , and the limit does not depend on the choice of the orthonormal basis. Therefore if T ∈ B 1 [X ] and {e γ } is any orthonormal basis for X , then set tr(T ) = γ T e γ ; e γ and hence T 1 = tr |T | .
The number tr(T ) ∈ F is the trace of T ∈ B 1 [X ] (so the terminology trace-class).
The norm · 1 = tr | · | on the linear space B 1 [X ] of all trace-class operators is referred to as the trace norm. Thus the trace-class itself can be written as Equivalently, for every S, T ∈ B 2 [X ] and for an arbitrary orthonormal basis {e γ }, Since tr(·) is linear, tr(T * ) = tr(T ), and T ; T 2 = tr(T * T ) = T 2 2 , the function · ; · 2 is a Hermitian symmetric sesquilinear form which induces a quadratic form. In other words, · ; · 2 is an inner product on B 2 [X ] that induces the norm · 2 : (B 2 [X ], · 2 ) is an inner product space where the norm · 2 is generated by the inner product defined by T ; S 2 = tr(S * T ) for every S, T ∈ B 2 [X ].
Note that the Schwartz inequality for this inner product on B 2 [X ] is a straightforward consequence of the definition of the Hilbert-Schmidt norm · 2 (and, of course, of the Schwartz inequality on both Hilbert spaces, X and ℓ 2 over Γ): Similarly to Theorem 5.5, it can be verified that (a) (B 2 [X ], · 2 ) is a Hilbert space, . In fact, a proof of (a) follows the same argument as in the proof of Theorem 5.5(a) with B 1 [X ] and · 1 replaced by B 2 [X ] and · 2 . This shows that (B 2 [X ], · 2 ) is complete, thus a Banach space, and so a Hilbert space since the norm · 2 is induced by an inner product: T 2 = T ; T 2 1 2 = tr |T | 2 1 2 for every T ∈ B 2 [X ]. In the same way, a proof of (b) follows exactly as the proof of Theorem 5.5(b).

Conclusion
Quite often the term nuclear operator is tacitly attributed to trace class operators without further explanation such as, for instance, "an operator will be called nuclear if it belongs to B 1 [X ]" [7, Section III.8]. This prompts our final result. for any orthonormal basis {e k } for H. So T 1 = min k |α k |, the minimum taken over all scalar summable sequences {α k } as in the above representation of T .
From (a) and (b) we get the following statement: An operator T lies in B 1 [X ] if and only if there are X -valued unit sequences {z k } and {y k } (i.e., z k = y k = 1) and a scalar summable sequence {α k } (i.e., k |α k | < ∞) such that T x = k α k x ; z k y k for every x in X . Moreover, T 1 = inf k |α k ], where the infimum is taken over all scalar summable sequences for which the above expression for T x holds.
Such an expression for T x is precisely a nuclear representation of T as in Section 4. Then B 1 [X ] = B N [X ]. Also, with the infimum taken over all nuclear representations of T ∈ B N [X ], we get (for arbitrary unit sequences {z k } and {y k } and summable scalar sequence {α k }) T N = inf k |α k | z k y k = inf k |α k | = T 1 .
Thus the notions of nuclear and trace-class coincide on Hilbert spaces. For their relationship beyond Hilbert spaces, the same first lines of Bartle's review in Mathematical Reviews we borrowed to open the paper can be used to close it, viz., "Grothendieck showed that a Banach space X has the approximation property if and only if for every nuclear operator T = k f k (·)y k the number tr(T ) = k f k (y k ) is well-defined" (see, e.g., [4, Theorems 1.3.6, 1.3.11, 1.4.18 and Proposition 1. 4.19]). This can be regarded as a starting point for characterizing the trace property in Banach spaces. For further readings along this line see, for instance, [21,11,6].