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Some Banach Spaces Are Almost Hilbert 3 . SOME BANACH SPACES ARE ALMOST HILBERT TEPPER L. GILL1∗ AND MARZETT GOLDEN1 Abstract. The purpose of this note is to show that, if is a uniformly convex Banach, then the dual space ′ has a “ Hilbert space representation”B (defined in the paper), that makes Bmuch closer to a Hilbert space then previously suspected. As an application,B we prove that, if also has a Schauder basis (S- basis), then for each A C[ ] (the closed and denselyB defined linear operators), there exists a closed densely∈ B defined linear operator A∗ C[ ] that has all the expected properties of an adjoint. Thus for example,∈ theB bounded linear operators, L[ ], is a ∗algebra. This result allows us to give a natural definition to the SchattenB class of operators on a uniformly convex Banach space with a S-basis. In particular, every theorem that is true for the Schatten class on a Hilbert space, is also true on such a space. The main tool we use is a special version of a result due to Kuelbs [K], which shows that every uniformly convex Banach space with a S-basis can be densely and continuously embedded into a Hilbert space which is unique up to a change of basis. 1. Introduction In 1965, Gross [G] proved that every real separable Banach space contains a separable Hilbert space as a dense embedding, and this space is the support of a Gaussian measure. This was a generalization of Wiener’s theory, that was based on the use of the (densely embedded Hilbert) Sobolev space H1[0, 1] C[0, 1]. In 1972, Kuelbs [K] generalized Gross’ theorem to include the Hilbert space⊂ rigging H1[0, 1] C[0, 1] L2[0, 1]. This general theorem can be stated as: ⊂ ⊂ Theorem 1.1. (Gross-Kuelbs)Let be a separable Banach space. Then there exist separable Hilbert spaces , B and a positive trace class operator T de- H1 H2 12 fined on 2 such that 1 2 all as continuous dense embeddings, with 1/2 H1/2 H ⊂B⊂H1/2 1/2 T12 u, T12 v =(u, v)2 and T12− u, T12− v =(u, v)1. 1 2 arXiv:1507.08611v1 [math.FA] 27 Jul 2015 This theorem makes it possible to give a definition of the adjoint for bounded linear operators on separable Banach spaces. The definition has all the expected properties. In particular, It can be shown that, for each bounded linear operator A on , there exists A∗, with A∗A, maximal accretive, selfadjoint, (A∗A)∗ = A∗A, B and I + A∗A is invertible (see [GBZS]). The basic idea is simple, let A be bounded on and let A be the restriction B 1 of A to 1. We can now consider A1 : 1 2. If J2 : 2 2′ is the standard H H → H H 1→ H conjugate isomorphism, then (A′ )J : ′ , so that J− (A′ )J : 1 2 H1 → H2 1 1 2 H1 → H1 ⊂ Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. ∗ Corresponding author. 2010 Mathematics Subject Classification. Primary 47B37; Secondary 46B10; 46C99. Key words and phrases. adjoint on Banach space, Lax Theorem; Schatten Classes. 1 2 T.L.GILLANDM.GOLDEN 1 1 . It follows that J1− (A1′ )J2 : . It easy to show that A∗ = J1− (A1′ )J2 hasB 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 <p< , the family of vectors B ∞ 1, cos(2πt), sin(2πt) cos(4πt), sin(4πt),... { } is a norm one S-basis for . B SOME BANACH SPACES ARE ALMOST HILBERT 3 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 <p< , then the standard example is ∈ ∈ B ∞ 2 p p 2 q 1 1 u∗ = (u)(x)= u − u(x) − u(x) L [Ω], + =1. J k kp | | ∈ p q Furthermore, 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 .
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