Regularized Trace for Operators on a Separable Banach Space

In this paper we consider a Sturni-Liouville type differential operator with unbounded operator coefficients given on a finite interval, with values in a separable Banach space B. In the past, problems of this type have been mainly studied on Hilbert space. Kuelbs (J Funct Anal 5:354-367, 1970) has shown that every separable Banach space B can be continuously embedded in a separable Hillbert space H. Given this result, we first prove that there always exists a separable Banach space B-z* subset of H* as a continuous embedding, which is a (conjugate) isometric isomorphic copy of B. This space generates a semi-inner product structure for B and is the tool we use to develop our theory. We are able to obtain a regularized trace formula for the above differential operator when the problem is posed on B. We also provide a few examples illustrating the scope and implications of our approach.