312,054 research outputs found
Spectral Invariance of Non-Smooth Pseudodifferential Operators
In this paper we discuss some spectral invariance results for non-smooth
pseudodifferential operators with coefficients in H\"older spaces. In analogy
to the proof in the smooth case of Beals and Ueberberg, we use the
characterization of non-smooth pseudodifferential operators to get such a
result. The main new difficulties are the limited mapping properties of
pseudodifferential operators with non-smooth symbols and the fact, that in
general the composition of two non-smooth pseudodifferential operators is not a
pseudodifferential operator.
In order to improve these spectral invariance results for certain subsets of
non-smooth pseudodifferential operators with coefficients in H\"older spaces,
we improve the characterization of non-smooth pseudodifferential operators in a
previous work by the authors.Comment: 43 page
Very smooth points of spaces of operators
In this paper we study very smooth points of Banach spaces with special
emphasis on spaces of operators. We show that when the space of compact
operators is an -ideal in the space of bounded operators, a very smooth
operator attains its norm at a unique vector (up to a constant
multiple) and is a very smooth point of the range space. We show that if
for every equivalent norm on a Banach space, the dual unit ball has a very
smooth point then the space has the Radon--Nikod\'{y}m property. We give an
example of a smooth Banach space without any very smooth points.Comment: 12 pages, no figures, no table
Operators with smooth functional calculi
We introduce a class of (tuples of commuting) unbounded operators on a Banach
space, admitting smooth functional calculi, that contains all operators of
Helffer-Sj\"ostrand type and is closed under the action of smooth proper
mappings. Moreover, the class is closed under tensor product of commuting
operators. In general an operator in this class has no resolvent in the usual
sense so the spectrum must be defined in terms of the functional calculus. We
also consider invariant subspaces and spectral decompositions
On the Schatten-von Neumann properties of some pseudo-differential operators
We obtain a number of explicit estimates for quasi-norms of
pseudo-differential operators in the Schatten-von Neumann classes with
. The estimates are applied to derive semi-classical bounds for
operators with smooth or non-smooth symbols.Comment: 22 page
Functional calculus for non-commuting operators with real spectra via an iterated Cauchy formula
We define a smooth functional calculus for a non-commuting tuple of
(unbounded) operators on a Banach space with real spectra and resolvents
with temperate growth, by means of an iterated Cauchy formula. The construction
is also extended to tuples of more general operators allowing smooth functional
calculii. We also discuss the relation to the case with commuting operators
Satellite operators as group actions on knot concordance
Any knot in a solid torus, called a pattern or satellite operator, acts on
knots in the 3-sphere via the satellite construction. We introduce a
generalization of satellite operators which form a group (unlike traditional
satellite operators), modulo a generalization of concordance. This group has an
action on the set of knots in homology spheres, using which we recover the
recent result of Cochran and the authors that satellite operators with strong
winding number give injective functions on topological concordance
classes of knots, as well as smooth concordance classes of knots modulo the
smooth 4--dimensional Poincare Conjecture. The notion of generalized satellite
operators yields a characterization of surjective satellite operators, as well
as a sufficient condition for a satellite operator to have an inverse. As a
consequence, we are able to construct infinitely many non-trivial satellite
operators P such that there is a satellite operator for which
is concordant to K (topologically as well as smoothly in a
potentially exotic ) for all knots K; we show that these
satellite operators are distinct from all connected-sum operators, even up to
concordance, and that they induce bijective functions on topological
concordance classes of knots, as well as smooth concordance classes of knots
modulo the smooth 4--dimensional Poincare Conjecture.Comment: 20 pages, 9 figures; in the second version, we have added several new
results about surjectivity of satellite operators, and inverses of satellite
operators, and the exposition and structure of the paper have been improve
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