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1 1 . It follows that J1− (A1′ )J2 : . It easy to show that A∗ = J1− (A1′ )J2 Bhas the the main properties of|B anB adjoint → B for A on . |B At this level of generality, the definition of anB adjoint for closed operators depends on the domain of the operator and changes the choice of 1 (and T12) for each operator. Thus, the adjoint is only reasonable for boundedH operators. It is a open question if all closed densely defined linear operators can have an adjoint with all the expected properties. Part of the problem is that not all operators in [ ] are of Baire class one (for example, when is nonreflexive). An operator A C B B is of Baire class one if and only if it can be approximated by a sequence, An , of bounded linear operators. The solution is unknown and we suspect that,{ in} general, there may be at least one operator in [ ] without an adjoint. C B 1.1. Purpose. In this note, we focus on uniformly convex Banach spaces, the best class of spaces that are not Hilbert. Our purpose is to show that these spaces are very close to Hilbert spaces and give the best possible results. In this case, the only difference between the bounded linear operators L[ ] and L[ ] is that B H L[ ] is not a C∗-algebra. Our main tool is a new representation for the dual space.B We embed into a (single) Hilbert space that allows us to define an B H adjoint A∗ on for each closed densely defined linear operator A. We are also able to define aB natural Schatten class structure for L[ ], that is almost identical to the Schatten class on . B H 1.2. Preliminaries. The following theorem is due to Lax [L]. Theorem 1.2. (Lax’s Theorem) Let be a separable Banach space that is con- tinuously and densely embedded in aB Hilbert space and let T be a bounded linear operator on that is symmetric with respectH to the inner product of (i.e., (Tu,v) =(u,TvB ) for all u, v ). Then: H H H ∈ B (1) The operator T is bounded with respect to the norm and H 2 2 T ∗T = T 6 k T , k kH k kH k kB where k is a positive constant. (2) The spectrum of T relative to is a subset of the spectrum of T relative to . H (3) TheB point spectrum of T relative to is a equal to the point spectrum of T relative to . H B Definition 1.3. A family of vectors in a Banach space , is called a {En} ⊂ B Schauder basis (S-basis) for if n = 1 and, for each u , there is a B kE kB ∈ B unique sequence (un) of scalars such that k ∞ u = limk un n = un n. →∞ E E n=1 n=1 X X For example, if = Lp[0, 1], 1 Definition 1.4. A duality map : ′, is a set J B 7→ B 2 2 (u)= u∗ ′ u,u∗ = u = u∗ ′ , u . J ∈ B h i k kB k kB ∀ ∈ B If is uniformly convex, (u ) contains a unique functional u∗ ′ for each u B . J ∈ B ∈ B Let Ω be a bounded open subset of Rn, n N. If u Lp[Ω] = , 1 2 p p 2 2 u,u∗ = u − u(x) dλ (x)= u = u∗ (1.1) h i k kp | | n k kp k kq ZΩ It can be shown that is uniformly convex and that u∗ = (u) is uniquely defined B p J for each u . Thus, if un is an S-basis for L [Ω], then, when normalized, the ∈ B { } q p family vectors u∗ is an S-basis for L [Ω] = (L [Ω])′. The relationship between { n} u and u∗ is nonlinear. In the next section we prove the remarkable result, that there is another representation of ′, with u∗ = J (u) linear, for each u . B B ∈ B (However, u∗ is no longer a duality mapping.) 2. The Natural Hilbert space for a Uniformly Convex Banach Space In this section we construct the natural Hilbert space for a uniformly convex Banach space with an S-basis. (For this, we only need the Kuelbs part of The- orem 1.1.) Fix and let n be a S-basis for . For each n, let n∗ be the B {E } nB E E corresponding dual vector in ′ and set tn = 2− . For each pair u, v define a inner product: B ∈ B ∞ (u, v)= t ∗,u , v n hEn i hEn∗ i n=1 X and let be the completion of in the induced norm. Thus, densely and H B B ⊂ H 1/2 ∞ 2 u = tn n∗,u sup n∗,u sup ∗,u = u , (2.1) ∗ k kH |hE i| ≤ n |hE i| ≤ ′ |hE i| k kB "n=1 # B 1 X kE k ≤ so that the embedding is both dense and continuous. (It is clear that is unique up to a change of S-basis.) H 2.1. The Hilbert Space Representation. In this section, we show that the dual space of a uniformly convex Banach space has a “Hilbert space representa- tion”. Definition 2.1. If be a Banach space, we say that ′ has a Hilbert space representation if thereB exists a Hilbert space , with B as a continuous H B ⊂ H dense embedding and for each u′ ′, u′ =( ,u) for some u . ∈ B · H ∈ B Theorem 2.2. If be a uniformly convex Banach space with an S-basis, then B ′ has a Hilbert space representation. B 4 T.L.GILLANDM.GOLDEN Proof. Let be the natural Hilbert space for and let J be the natural linear H B mapping from ′, defined by H → H v, J(u) =(v,u) , for all u, v . h i H ∈ H It is easy to see that J is bijective and J∗ = J. First, we note that the restriction of J to , J , maps to a unique subset of linear functionals J (u), u and, J (Bu +Bv)= J (Bu)+ J (v), for each u, v . We are done{ ifB we can prove∈ B} B B B ∈ B that J (u), u = ′. For this, it suffices to show that J (u) is bounded for each {u B . Since∈ B} is denseB in , from equation (2.1) we have:B ∈ B B H v, J (u) v, J (u) J (u) ′ = sup h B i 6 sup h B i = u 6 u . k B kB v v v v k kH k kB ∈B k kB ∈B k kH Thus, J (u), u ′. Since is uniformly convex, J (u), u = { B ∈B}⊂B B { B ∈ B} ′. B 2.2. Construction of the adjoint on . We can now show that each closed densely linear operator on has a naturalB adjoint defined on . B B Theorem 2.3. Let be a uniformly convex Banach space with an S-basis. If C[ ] denotes the closed denselyB linear operators on and L[ ] denotes the boundedB B B linear operators, then every A [ ] has a well defined adjoint A∗ [ ]. ∈ C B ∈ C B Furthermore, if A L[ ], then A∗ L[ ] with: ∈ B ∈ B (1) (aA)∗ =¯aA∗, (2) A∗∗ = A, (3) (A∗ + B∗)= A∗ + B∗ (4) (AB)∗ = B∗A∗ and 2 (5) A∗A A . k kB ≤k kB Thus, L[ ] is a ∗algebra. B Proof. First, let J be the natural linear mapping from ′ and let J be the H → H B restriction of J to . If A [ ], then A′J : ′ ′. Since A′ is closed and B ∈ C B 1 B B → B densely defined, it follows that J− A′J : is a closed and densely defined B 1 B B → B 1 linear operator. We define A∗ = [J− A′J ] [ ]. If A L[ ], A∗ = J− A′J B B ∈ C B ∈ B B B is defined on all of . By the Closed Graph Theorem, A∗ L[ ]. The proofs of (1)-(3) are straightB forward. To prove (4), ∈ B 1 1 (BA)∗ = J− (BA)′J = J− A′B′J B B B B 1 1 (2.2) = J− A′J J− B′J = A∗B∗. B B B B If we replace B by A∗ in equation (2.2), noting that A∗∗ = A, we also see that (A∗A)∗ = A∗A. To prove (5), we first see that: A∗Av, J (u) =(A∗Av,u) =(v, A∗Au) , h B i H H so that A∗A is symmetric. Thus, by Lax’s Theorem, A∗A has a bounded extension to and A∗A 6 k A∗A , where k is a positive constant. We also have that H k kH k kB 6 6 2 A∗A A∗ A A B . (2.3) k kB k kBk kB k k SOME BANACH SPACES ARE ALMOST HILBERT 5 2 It follows that A∗A A . If equality holds in (2.3), for all A L[ ], then k kB ≤k kB ∈ B it is a C∗-algebra. It is well-known that this is true if and only if is a Hilbert space. Thus, in general the inequality in (2.3) is strict. B 2.3. Operators on . For the remainder of the paper, we assume that is B B uniformly convex and ′ carries its Hilbert space representation. B Definition 2.4. Let U be bounded, A C[ ] and let , be subspaces of . Then: ∈ B U V B (1) A is said to be naturally self-adjoint if D(A)= D(A∗) and A = A∗. (2) A is said to be normal if D(A)= D(A∗) and AA∗ = A∗A. (3) U is unitary if UU ∗ = U ∗U = I. (4) The subspace is to if and only, for each v and u , (v,u) =0 Uand, for⊥ eachV u and v , (u, v) ∈=0 V. ∀ ∈ U H ∈ U ∀ ∈V H The last definition is transparent since, orthogonal subspaces in induce or- thogonal subspaces in . H B Theorem 2.5. (Gram-Schmidt) For each fixed basis ϕ , 1 6 i< of , there { i ∞} B is at least one set of dual functionals Si such that ψi , Si , 1 6 i< is a biorthonomal set of vectors for , (i.e.,{ } ψ ,S = δ{{). } { } ∞} B h i ji ij Proof. Since each ϕ is in , we can construct an orthogonal set of vectors i H φi, 1 6 i< in by the standard Gram-Schmidt process. Set ψi = φi/ φi { ∞} H k kB and let ψ∗ = J(ψ). From here, it is easy to check that ψ , ψ∗ , 1 6 i< i {{ i} { i } ∞} is a biorthonormal set and the family ψ∗ is unique. { i } We close this section with the following observation about the use of . Let A be any closed densely defined naturally selfadjoint linear operator on H with B a discrete spectrum λi . It can be extended to with the same properties. If { } ∗ H ¯ Aψ,ψ (Aψ,ψ)H we compute the ratio h ∗ i in , it will be “close” to the value of in ψ,ψ B (ψ,ψ)H . By Lax’s Theorem [hL], thei extension from to does not change the point Hspectrum, so we can use the min-max theoremB on Hto compute the eigenvalues and eigenfunctions of A via A¯ exactly. Since is denseH in , it follows that the min-max theorem also holds on . B H B 2.3.1. Selfadjointness. With respect to our definition of natural selfadjointness, the following related definition is due to Palmer [P], where the operator is called symmetric. This is essentially the same as a Hermitian operator as defined by Lumer [LU]. (An operator A is dissipative if A is accretive.) − Definition 2.6. A closed densely defined linear operator A on is called self- conjugate if both iA and iA are dissipative. B − Theorem 2.7. (Vidav-Palmer) A linear operator A, defined on , is self- conjugate if and only if iA and iA are generators of isometric semigroups.B − Theorem 2.8. The operator A, defined on , is self-conjugate if and only if it is naturally self-adjoint. B 6 T.L.GILLANDM.GOLDEN Proof. Let A¯ and A¯∗ be the closed densely defined extensions of A and A∗ to . On , A¯ is naturally self-adjoint if and only if iA¯ generates a unitary group,H if andH only if it is self-conjugate. Thus, both definitions coincide on . It follows that the restrictions coincide on . H B The proof of the last theorem represents a general approach for proving new results for . The following are two representative. B Theorem 2.9. (Polar Representation) Let be a uniformly convex Banach space with an S-basis. If A C[ ], then there existsB a partial isometry U and a naturally self-adjoint operator ∈T , withB D(T ) = D(A) and A = UT . Furthermore, T = 1/2 [A∗A] , in a well-defined sense. Theorem 2.10. (Spectral Representation) Let be a uniformly convex Banach space with an S-basis and let A C[ ], be a naturallyB self-adjoint linear operator. ∈ B Then, there exists a operator-valued spectral measure Ex, x R and for each u D(A), ∈ ∈ Au = xdEx(u). ZR 3. Examples The following related Hilbert space is more general. It is a concrete implemen- tation of the abstract construction of Kuelbs [K] and, in a different form, is due to Steadman [ST]. It is constructed over L1, which is not uniformly convex, but is more suitable for applications. It was first used to provide a rigorous foundation for the Feynman path integral formulation of quantum mechanics in [GZ]. We use it in this section to provide a few concrete examples. 2 n n n n 3.1. The space KS [R ]. On R let Q be the set x =(x1, x2 , xn) R { ··· ∈ n} such that xi is rational for each i. Since this is a countable dense set in R , n 1 2 3 i we can arrange it as Q = x , x , x , . For each l and i, let Bl(x ) be the closed cube centered at xi, with{ sides parallel···} to the coordinate axes and diagonal l rl = 2− ,l N. Now choose the natural order which maps N N bijectively to N: ∈ × (1, 1), (2, 1), (1, 2), (1, 3), (2, 2), (3, 1), (3, 2), (2, 3), . { ···} i Let Bk, k N be the resulting set of (all) closed cubes Bl(x ) (l, i) N N { ∈ } n { | ∈ × } centered at a point in Q and let k(x) be the characteristic function of Bk, so that (x) is in Lp[Rn] for 1 p E . Define F ( ) on L1[Rn] by Ek ≤ ≤∞ k · Fk(f)= k(x)f(x)dλn(x). (3.1) n E ZR p n It is clear that Fk( ) is a bounded linear functional on L [R ] for each k, · p n Fk 1 and, if Fk(f) = 0 for all k, f = 0 so that Fk is fundamental on L [R ] k k ≤ k { } for 1 p . Set tk = 2− , so that k∞=1 tk = 1 and define a measure dµ on Rn ≤Rn by:≤ ∞ × P ∞ dµ = tk k(x) k(y) dλn(x)dλn(y). k=1 E E hX i SOME BANACH SPACES ARE ALMOST HILBERT 7 To construct our Hilbert space, define an inner product ( ) on L1[Rn] by · (f,g)= f(x)g(y)∗dµ Rn Rn Z × (3.2) ∗ ∞ = tk k(x)f(x)dλn(x) k(y)g(y)dλn(y) . k=1 Rn E Rn E X Z Z We call the completion of L1[Rn], with the above inner product, the Kuelbs- Steadman space, KS2[Rn]. Steadman [ST] constructed this space by modifying a method developed by Kuelbs [K] for other purposes. Her interest was in showing that L1[R] can be densely and continuously embedded in a Hilbert space which contains the non-absolutely integrable functions. To see that this is the case, suppose f is a non-absolutely integrable function, say Henstock-Kurzweil integral (or of any other type see [H]), then: 2 2 2 ∞ f KS = tk k(x)f(x)dλn(x) k k k=1 n E ZR X 2 6 sup k(x)f(x )dλn(x) < . n E ∞ k ZR Since the absolute value is outside the integral, we see that f KS2[Rn] for any of the definitions of a non-absolute integral (see [GO]). A detailed∈ discussion of this space and its relationship to the Feynman path integral formulation to quantum mechanics, can be found in [GZ] Theorem 3.1. The space KS2[Rn] contains Lp[Rn] (for each p, 1 6 p 6 ) as dense subspaces. ∞ Proof. By construction, we know that KS2[Rn] contains L1[Rn] densely. Thus, we need only show that Lq[Rn] KS2[Rn] for q = 1. Let f Lq[Rn] and q < . Since (x) = (x) 6 1 and (⊂x) q 6 (x), we6 have ∈ ∞ |E | E |E | E 2q 1/2 q 2 ∞ f KS = tk k(x)f(x)dλn(x) k k k=1 n E " ZR # X 2 1/2 q ∞ q 6 tk k(x) f(x) dλn(x) k=1 n E | | " ZR # X 1 q q 6 sup k(x) f(x) dλn(x) 6 f q . n E | | k k k ZR 2n Hence, f KS2[Rn]. For q = , first note that vol(B )2 1 , so we have ∈ ∞ k ≤ 2√n h i 2 1/2 2 ∞ f KS = tk k(x)f(x)dλn(x) k k k=1 n E " R # X Z 1/2 n ∞ 2 2 1 6 tk[vol(B k)] [ess sup f ] 6 f . k=1 | | 2√n k k∞ hhX i i 8 T.L.GILLANDM.GOLDEN 2 n n 2 n Thus f KS [R ], and L∞[R ] KS [R ]. ∈ ⊂ n 2 n 2 n The fact that L∞[R ] KS [R ], while KS [R ] is separable makes it clear in a very forceful manner⊂ that separability is not an inherited property. We note that, since L1[Rn] KS2[Rn] and KS2[Rn] is reflexive, the second dual 2⊂ L1[Rn]′′ = M[Rn] KS [Rn]. Recall that M[Rn] is the space of bounded finitely additive set functions⊂ defined on the Borel sets B[Rn]. This space contains the Dirac delta measure and free-particle Green’s function for the Feynman path integral. The next result is an unexpected benefit. p Theorem 3.2. Let fn f weakly in L , 1 p , then fn f strongly in KS2 (i.e., every weakly→ compact subset of Lp≤is compact≤ ∞ in KS2→). Proof. The proof of follows from the fact that, if fn is any weakly convergent sequence in Lp with limit f, then { } k(x)[fn(x) f(x)] dλn(x) 0 n E − → ZR for each k. It follows that f converges strongly to f in KS2. { n} Let A be a closed densely defined linear operator defined on Lp[Rn], 1 3.1.1. Example: Integral Operators. In one dimension, the Hilbert transform can be defined on L2[R] via its Fourier transform: H[(f)= i sgn x f.ˆ − It can also be defined directly as principal-value integral: 1 f(y) (Hf)(x) = lim dy. ε 0 π x y >ε x y → Z| − | − For a proof of the following results see Grafakos [GRA], chapter 4. Theorem 3.5. The Hilbert transform on L2[R] satisfies: (1) H is an isometry, H(f) 2 = f 2 and H∗ = H. (2) For f Lp[R], 1 n 1 Ω(y)dσ(y)=0, where dσ is the induced Lebesgue measure on S − . n−1 ZS (3) The function Ω(x) is said to have the Dini-type condition if 1 ω(δ)dδ sup Ω(x) Ω(y) 6 ω(δ) < . x y 6δ | − | ⇒ 0 δ ∞ | − | Z x = y =1 | | | | A proof of the following theorem can be found in Stein [STE] (see pg., 39). Theorem 3.8. Suppose that Ω is homogeneous of degree 0, satisfying both the cancellation property and the Dini-type condition. If f Lp[Rn], 1 (1) There exists a constant Ap, independent of both f and ε such that T (f) 6 A f . k ε kp pk kp 10 T.L.GILLANDM.GOLDEN p (2) Furthermore, lim Tε(f)= T (f) exists in the L norm and ε 0 → T (f) 6 A f . (3.4) k kp pk kp Treating Tε(f) as a special case of the Henstock-Kurzweil integral, conditions (1) and (2) are automatically satisfied and we can write the integral as Ω(y x) T (f)(x)= − n f(y)dy. n y x ZR | − | q 1 1 For g L , + = 1, we have T (f),g = f, T ∗(g) . Using Fubini’s Theorem ∈ p q h i h i for the Henstock-Kurzweil integral (see [H]), we have that p Corollary 3.9. The adjoint of T, T ∗ = T , is defined on L and satisfies equation (3.4) − It is easy to see that the Riesz transform is a special case of the above Theorem and Corollary. Another closely related integral operator is the Riesz potential, Iα(f)(x) = α/2 p n ( ∆)− f(x), 0 <α On the other hand, it is shown in part I of Dunford and Schwartz [DS] that, if (Ω, Σ,µ) a positive measure space, then for L1 (Ω, Σ,µ) we have M( ) N( ). We assume that is uniformly convex, with a S-basis. In operatorB ⊂ theoreticB language, our S-basisB assumption is that the set of compact operators on have the approximation property, namely that every compact operator can be approx-B imated by operators of finite rank. In this section we will show that the structure of L[ ] is almost identical to its associated space L[ ]. The difference is that B 2 H L[ ] is not a C∗-algebra (i.e., A∗A = A , A L[ ], is not true for all A). BLet A be a compact operatork on kB andk letkBA¯ be∈ itsB extension to . For each B H compact operator A¯, there exists an orthonormal basis ϕ¯n n > 1 , for such that { | } H ∞ A¯ = µn(A¯)( , ϕ¯n) U¯ϕ¯n. n=1 · 2 1/2 Where the µn are the eigenvaluesX of [A¯∗A¯] = A¯ , counted by multiplicity and in decreasing order, and U¯ is the partial isometry associated with the polar de- composition of A¯ = U¯ A¯ . Without loss, we can assume that the set of functions ϕ¯n n > 1 is contained in and ϕn n > 1 is the normalized version in . If { | } B { | } ¯ B Sp[ ] is the Schatten Class of order p in L[ ], it is well-known that, if A Sp[ ], itsH norm can be represented as: H ∈ H 1/p p/2 1/p ∞ p/2 ¯ H ¯∗ ¯ ¯∗ ¯ A p = Tr A A = A Aϕ¯n, ϕ¯n (n=1 H ) n o X 1/p ∞ p = µn A¯ . (n=1 ) X Definition 4.1. We define the Schatten Class of order p in L[ ] by: B Sp[ ]= Sp[ ] . B H |B Since A¯ is the extension of A S [ ], we can define A on by ∈ p B B ∞ A = µn(A) ,ϕn∗ Uϕn, n=1 h· i X where ϕn∗ is the unique functional in ′ associated with ϕn and U is the restriction of U¯ to . The corresponding normB of A on S [ ] is defined by: B p B 1/p ∞ p/2 A B = A∗Aϕn,ϕn∗ . k kp n=1 h i nX o Theorem 4.2. Let A Sp[ ], then A B = A¯ H. ∈ B k kp p Proof. It is clear that ϕ n > 1 is a set of eigenfunctions for A∗A on . Fur- { n | } B thermore, A∗A is naturally selfadjoint and compact, so that its spectrum is dis- crete. By Lax’s Theorem, the spectrum of A∗A is unchanged by its extension to 2 . It follows that A∗Aϕ = µ (A¯) ϕ , so that H n n n 2 2 A∗Aϕ ,ϕ∗ = µ (A) ϕ ,ϕ∗ = µ (A¯) , h n n i | n | h n ni n 12 T.L.GILLANDM.GOLDEN and 1/p 1/p p/2 p ∞ ∞ ¯ ¯ H A B = A∗Aϕn,ϕn∗ = µn(A) = A . k kp n=1 h i n=1 p nX o nX o It is clear that all of the theory of operator ideals on Hilbert spaces extend to uniformly convex Banach spaces with a S-basis in a straightforward way. We state a few of the more important results to give a sense of the power provided by the existence of adjoints for spaces of this type. The first result extends theorems due to Weyl [W], Horn [HO], Lalesco [LA] and Lidskii [LI]. The proofs are all straight forward, for a given A extend it to , use the Hilbert space result and then restrict back to . H B Theorem 4.3. Let A K( ), the set of compact operators on , and let λn be the eigenvalues of A∈countedB up to algebraic multiplicity. If Φ isB a mapping{ on} [0, ] which is nonnegative and monotone increasing, then we have: ∞ (1) (Weyl) ∞ ∞ Φ( λn(A) ) 6 Φ(µn(A)) n=1 | | n=1 and X X (2) (Horn) If A , A K( ) 1 2 ∈ B ∞ ∞ Φ( λn(A1A2) ) 6 Φ(µn(A1)µn(A2)). n=1 | | n=1 In case XA S1( ), we have: X (3) (Lalesco) ∈ B ∞ ∞ λn(A) 6 µn(A) n=1 | | n=1 and X X (4) (Lidskii) ∞ λn(A)= Tr(A). n=1 Simon [SI2] provides a veryX nice approach to infinite determinants and trace class operators on separable Hilbert spaces. He gives a comparative historical analysis of Fredholm theory, obtaining a new proof of Lidskii’s Theorem as a side benefit and some new insights. A review of his paper shows that much of it can be directly extended to operator theory on uniformly convex Banach spaces. 4.2. Discussion. On a Hilbert space , the Schatten classes S ( ) are the only H p H ideals in K( ), and S1( ) is minimal. In a general Banach space, this is far from true. A completeH historyH of the subject can be found in the recent book by Pietsch [PI1] (see also Retherford [R], for a nice review). We limit this discussion to a few major topics on the subject. First, Grothendieck [GR] defined an important class of nuclear operators as follows: Definition 4.4. If A F( ) (the operators of finite rank), define the ideal N1( ) by: ∈ B B N ( )= A F( ) N (A) < , 1 B { ∈ B | 1 ∞} SOME BANACH SPACES ARE ALMOST HILBERT 13 where m m N1(A) = glb fn φn fn ′, φn , A = φn , fn n=1 k kk k ∈ B ∈ B n=1 h· i n o and the greatest lowerX bound is over all possible representationsX for A. Grothendieck showed that N ( ) is the completion of the finite rank operators 1 B and is a Banach space with norm N1( ). It is also a two-sided ideal in K( ). It is easy to show that: · B Corollary 4.5. M( ), N( ) and N ( ) are two-sided *ideals. B B 1 B In order to compensate for the (apparent) lack of an adjoint for Banach spaces, Pietsch [PI2], [PI3] defined a number of classes of operator ideals for a given . Of particular importance for our discussion is the class C ( ), defined by B p B ∞ p Cp( )= A K( ) Cp(A)= [si(A)] < , B ∈ B i=1 ∞ n o where the singular numbers sn(A) are definedX by: sn(A) = inf A K rank of K 6 n . {k − kB | } Pietsch has shown that, C ( ) N ( ), while Johnson et al [JKMR] have 1 B ⊂ 1 B shown that for each A C ( ), ∞ λ (A) < . On the other hand, ∈ 1 B n=1 | n | ∞ Grothendieck [GR] has provided an example of an operator A in N1(L∞[0, 1]) with ∞ λ (A) = (see SimonP [SI1], pg. 118). Thus, it follows that, in n=1 | n | ∞ general, the containment is strict. It is known that, if C1( ) = N1( ), then is isomorphicP to a Hilbert space (see Johnson et al). It is clearB fromB the aboveB discussion, that: Corollary 4.6. C ( ) is a two-sided *ideal in K( ), and S ( ) N ( ). p B B 1 B ⊂ 1 B For a given Banach space, it is not clear how the spaces C ( ) of Pietsch relate p B to our Schatten Classes Sp( ) (clearly Sp( ) Cp( )). Thus, one question is that of the equality of S ( )B and C ( ). (WeB suspect⊆ B that S ( )= C ( ).) p B p B 1 B 1 B 5. Conclusion The most interesting aspect of this paper is the observation that the dual space of a Banach space can have more then one representation. It is well-known that a given Banach space , can have many equivalent norms that generate the same topology. However, theB geometric properties of the space depend on the norm used. We have shown that the properties of the linear operators on depend B on the family of linear functionals used to represent the dual space ′. This approach offers a interesting tool for a closer study of the structure ofB bounded linear operators on . B References [CA] N. L. Carothers, A Short Course on Banach Space Theory, London Math. 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Palmer, Unbounded normal operators on Banach spaces, Trans. Amer. Math. Sci. 133 (1968), 385-414. [PI1] A. Pietsch, History of Banach Spaces and Operator Theory, Birkh¨auser, Boston, (2007). [PI2] A. Pietsch, Einige neue Klassen von kompacter linear Abbildungen, Revue der Math. Pures et Appl. (Bucharest), 8 (1963), 423–447. [PI3] A. Pietsch, Eigenvalues and s-Numbers , Cambridge University Press, (1987). [R] J. R. Retherford, Applications of Banach ideals of operators, Bulletin of the American Mathematical Society, 81 (1975), 978-1012. [SI1] B. Simon, Trace Ideals and their Applications, London Mathematical Society Lecture Notes Series 35, Cambridge University Press, New York, (1979). [SI2] B. Simon, Notes on Infinite Determinants of Hilbert Space Operators, Advances in math. 24, (1977) 244-273. [SC] R. Schatten, A Theory of Cross-Spaces, Princeton University Press, Princeton , New Jersey, (1950). [ST] V. Steadman, Theory of operators on Banach spaces, Ph.D thesis, Howard University, (1988). [STE] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton University Press, Princeton, NJ, (1970). [W] H. Weyl, Inequalities between the two kinds of eigenvalues of a linear transformation, Proc. Nat. Acad. Sci. 35 (1949), 408-411. 1 Department of Mathematics, Howard University, Washington DC 20059, USA. E-mail address: [email protected]; [email protected] SOME BANACH SPACES ARE ALMOST HILBERT 15 2 Department of Mathematics, Howard University, Washington DC 20059, USA. E-mail address: [email protected]