Regularized Trace for Operators on a Separable Banach Space

Abstract

In this paper we consider a Sturm–Liouville type differential operator with unbounded operator coefficients given on a finite interval, with values in a separable Banach space $\mathcal B$. 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 $\mathcal B$ can be continuously embedded in a separable Hillbert space $\mathcal H$. Given this result, we first prove that there always exists a separable Banach space $\mathcal B_z^* \subset \mathcal H^*$ as a continuous embedding, which is a (conjugate) isometric isomorphic copy of $\mathcal B$. This space generates a semi-inner product structure for $\mathcal B$ 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 $\mathcal B$. We also provide a few examples illustrating the scope and implications of our approach.