A note on tensor fields in Hilbert spaces

We discuss and extend to infinite dimensional Hilbert spaces a well-known tensoriality criterion for linear endomorphisms of the space of smooth vector fields in R.


INTRODUCTION
A basic fact in classical differential geometry, called the "Fundamental lemma of differential geometry" in Besse (1987), is that an R-linear endomorphism of the space of vector fields in an open set of R n , is a tensor field (of type (1,1)) if and only if is linear with respect to functions.This is not any more true in infinite dimensions and the aim of this note is to give a contra-example and to introduce a class of endomorphism for which the criterion holds true.Some basic references for infinite dimensional differential geometry are Lang (1995) and Abraham et al. (1988).

TENSOR FIELDS IN HILBERT SPACES
Let H be a real Hilbert space, B(H) be the space of bounded linear endomorphisms of H and ⊂ H be an open set.We will denote by H( ) the space of (smooth) vector fields in , i.e., the space of smooth maps ξ : → H; by F( ) we denote the algebra of smooth real valued functions in .Then H( ) is a real vector space and an F( )-module in the obvious way.
We consider a map A : → B(H) and, for ξ ∈ H( ), we define a vector field Ã(ξ ) in by setting Ã(ξ )(x) = A(x) • ξ(x).If Ã(ξ ) is smooth for all ξ ∈ H( ) then we way that A is weakly 208 LEONARDO BILIOTTI, FRANCESCO MERCURI AND DANIEL V. TAUSK smooth; in this case Ã is a F( )-linear endomorphism of H( ).Clearly, if A is smooth then it is also weekly smooth and if dim(H) < +∞ the converse holds.Now we consider an arbitrary R-linear map Ã : H( ) → H( ).Recall that: 1. Ã is a tensor field (of type (1, 1)) if there exist a weakly smooth map A : → B(H) such that: and for every open subset U ⊂ .
Conditions ( 1) and ( 2) are obviously equivalent and any of them implies condition (3).It is well known and easy to prove that if dim(H) = n < +∞ then conditions (1), ( 2) and ( 3) are all equivalent.We briefly recall the proof of the implication (3)⇒(2).Let (e i ) n i=1 , n = dim(H) < +∞, be a basis for H and, given ξ ∈ H( ), write ξ = n i=1 ξ i e i , where each from which condition (2) follows.The aim of this note is to discuss the relations between conditions (2) and (3) in the case that H is infinite-dimensional.We start by pointing out that such conditions are no longer equivalent as the following example shows.
Example 2. Let λ : B(H) → H be a continuous non zero linear map whose kernel contains the closed subspace of compact endomorphisms of H. Set: where dξ(x) ∈ B(H) denotes the differential of the smooth map ξ at the point x.We claim that Ã is an F( )-linear operator.Namely, if f ∈ F( ) and if f (x) = 0 for some x ∈ then: Hence, d(f ξ )(x) ∈ B(H) has 1-dimensional range and therefore it is a compact operator and Ã(f ξ )(x) = 0. Now, in general, if f ∈ F( ) and x ∈ , we have: which proves that Ã is F( )-linear.On the other hand Ã cannot be punctual because if ξ ∈ B(H) is not in the kernel of λ then Ã(ξ )(0) is not zero.
If we assume the continuity of the operator Ã with respect to pointwise convergence in H( ) then we can proceed in analogy with the finite dimensional case to prove the implication (3)⇒(2).Such assumption, however, is too strong since in all interesting cases directional derivatives appear in the expression for Ã.Assume, for simplicity, that H is separable (the general case follow substituting sequences for nets).Fix a Hilbert space basis (e i ) +∞ i=1 for H then, for ξ ∈ H( ), we have ξ = +∞ i=1 ξ i e i , ξ i ∈ F( ), and this series converges pointwise together with all its directional derivatives.In fact, the i-th coordinate operator H ξ → ξ i = ξ, e i ∈ R is linear and continuous and hence it commutes with directional derivative operators The above considerations lead to the following: Proof.Let ξ ∈ H( ), x ∈ and assume ξ(x) = 0. We write ξ = +∞ i=1 ξ i e i , ξ i ∈ F( ), where (e i ) +∞ i=1 is a Hilbert basis for H.By setting ξ k = k i=1 ξ i e i then, as observed above, (ξ k ) k∈N converges pointwise to ξ together with all its directional derivatives.Since Ã is weakly which implies Ã(ξ )(x) = 0 and concludes the proof.
We will now discuss the result above in an interesting case.We recall that a linear connection in H is an R-bilinear map: • for all X, Y ∈ H( ), f ∈ F( ), the identity: holds.
An easy modification of Example 2 shows that not all connections are punctual in the first argument.However, if we give a Riemannian metric in i.e., a smooth map g : → B(H) such that for all x ∈ , g(x) : H → H is self-adjoint and there exists a positive real-valued function K such that g(x)v ≥ K(x) v , we have a special connection, the Levi-Civita connection, uniquely defined by the two extra conditions: