Eurasian Math. J., 2021 Volume 12, Number 3,Pages 78–89(Mi emj416)
Stokes-type integral equalities for scalarly essentially integrable locally convex vector-valued forms which are functions of an unbounded spectral operator
Abstract:
In this work we establish a Stokes-type integral equality for scalarly essentially integrable forms on an orientable smooth manifold with values in the locally convex linear space
$\langle B(G),\sigma(B(G),\mathcal{N})\rangle$, where $G$ is a complex Banach space and $\mathcal{N}$ is a suitable linear subspace of the norm dual of $B(G)$. This result widely extends the Newton-Leibnitz-type equality stated in one of
our previous articles. To obtain our equality we generalize the main result of those articles, and
employ the Stokes theorem for smooth locally convex vector-valued forms established there. Two
facts are remarkable. First, the forms integrated involved in the equality are functions of a possibly
unbounded scalar-type spectral operator in $G$. Secondly, these forms need not be smooth nor even
continuously differentiable.
Keywords and phrases:unbounded spectral operators in Banach spaces, functional calculus, integration of locally convex vector-valued forms on manifolds, Stokes equalities.
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