Some properties of Hilbert scales

Abstract

1. Hilbert scales. Suppose Eo is a separable complex Hilbert space with inner product (, )o. Let A: E0-*Eo be a completely continuous linear mapping with A >0. Then A can be represented in the form Au= Ej'Xj(u, aj)0aj where {aj} is a complete orthonormal set in Eo, Xl1X2?> . . . >0, and lim Xj=0, (see [1, Chapter 1]). Let E=noo? o AnEo and for u, vEE, aCER define (u, v)a= (A-au, A-av)o. Denote by Ea, E equipped with the a= (a )1/2 topology, and let Ea denote the completion. It follows that E=naEa. The family a-*Ea is called the Hilbert scale defined by A, and E is called the center (see [2, p. 93]). We will suppose that E carries the weakest topology in which all inclusions E--*Ea are continuous. It follows that E is a perfect space in the sense of [1 ] and VaEa = E* is its dual space. Since ||u a= |A'u| a+: for all uECE and a, 1ECR it follows that AO extends to an isometry from Ea to Ea+0, which we will also denote by AO. Also, it follows from the definition that {Xjaj} is a complete orthonormal set in Ea and (u, aj)aaX2a = (u, aj)0 for uCEa, a> 0. An alternative method of introducing Hilbert scales follows from the observation that if i: Ei-*Eo denotes the inclusion map, then A = (i*i)112. Hence, suppose Ho and H1 are complex Hilbert spaces, H1 is dense in Hog and the inclusion map i: Hi->Ho is completely continuous. Then i has the form u = i(u) = 0>= aj(u, bj)laj where lim aj =0, and { aj } and { bj } are bases in Ho and H1 respectively. It follows that i*i G 2 (H1, H1) has the form i*i(u) = E I ai | 2 (U, bj)jbj which extends to Ho by setting i*i(v) = E |a 24 2(v, aj)oaj. The pair {Hog (i*i) 12} then defines a Hilbert scale as above. An important class of Hilbert scales are those whose centers are nuclear spaces. It is not difficult to show that if a--*E, is a Hilbert scale defined by A, then the center is nuclear iff =i X' 0. Two simple examples are given below, and other examples can be found in [2] and [5]. (a) Suppose E denotes the set of all C? mappings u: R-*C having period one, and for u, vEE define (u, v)o=fJsu(t)i(t)dt and (u, v), = (u, v)o+ (Du, Dv)o. Let Ei denote the completion of E with respect to (, )1/2. Then the inclusion map i: El-*Eo is completely continuous