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The Resolvent Average for Positive Semidefinite Matrices
We define a new average - termed the resolvent average - for positive
semidefinite matrices. For positive definite matrices, the resolvent average
enjoys self-duality and it interpolates between the harmonic and the arithmetic
averages, which it approaches when taking appropriate limits. We compare the
resolvent average to the geometric mean. Some applications to matrix functions
are also given
Operators in Rigged Hilbert spaces: some spectral properties
A notion of resolvent set for an operator acting in a rigged Hilbert space
\D \subset \H\subset \D^\times is proposed. This set depends on a family of
intermediate locally convex spaces living between \D and \D^\times, called
interspaces. Some properties of the resolvent set and of the corresponding
multivalued resolvent function are derived and some examples are discussed.Comment: 29 page
A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains
In this note we investigate the asymptotic behaviour of the -numbers of
the resolvent difference of two generalized self-adjoint, maximal dissipative
or maximal accumulative Robin Laplacians on a bounded domain with
smooth boundary . For this we apply the recently introduced
abstract notion of quasi boundary triples and Weyl functions from extension
theory of symmetric operators together with Krein type resolvent formulae and
well-known eigenvalue asymptotics of the Laplace-Beltrami operator on
. It will be shown that the resolvent difference of two
generalized Robin Laplacians belongs to the Schatten-von Neumann class of any
order for which . Moreover, we also give a simple
sufficient condition for the resolvent difference of two generalized Robin
Laplacians to belong to a Schatten-von Neumann class of arbitrary small order.
Our results extend and complement classical theorems due to M.Sh.Birman on
Schatten-von Neumann properties of the resolvent differences of Dirichlet,
Neumann and self-adjoint Robin Laplacians
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