1,356 research outputs found
Multiplication Operators between Lipschitz-Type Spaces on a Tree
Let β be the space of complex-valued functions on the set of vertices of an infinite tree rooted at such that the difference of the values of at neighboring vertices remains bounded throughout the tree, and let β be the set of functions ββ such that |()β(β)|=(||β1), where || is the distance between and and β is the neighbor of closest to . In this paper, we characterize the bounded and the compact multiplication operators between β and β and provide operator norm and essential norm estimates. Furthermore, we characterize the bounded and compact multiplication operators between β and the space β of bounded functions on and determine their operator norm and their essential norm. We establish that there are no isometries among the multiplication operators between these spaces
Multiplication Operators on Weighted Banach Spaces of a Tree
We study multiplication operators on the weighted Banach spaces of an
infinite tree. We characterize the bounded and the compact operators, as well
as determine the operator norm. In addition, we determine the spectrum of the
bounded multiplication operators and characterize the isometries. Finally, we
study the multiplication operators between the weighted Banach spaces and the
Lipschitz space by characterizing the bounded and the compact operators,
determine estimates on the operator norm, and show there are no isometries
Spectral triples and finite summability on Cuntz-Krieger algebras
We produce a variety of odd bounded Fredholm modules and odd spectral triples
on Cuntz-Krieger algebras by means of realizing these algebras as "the algebra
of functions on a non-commutative space" coming from a sub shift of finite
type. We show that any odd -homology class can be represented by such an odd
bounded Fredholm module or odd spectral triple. The odd bounded Fredholm
modules that are constructed are finitely summable. The spectral triples are
-summable although their bounded transform, when constructed using the
sign-function, will already on the level of analytic -cycles be finitely
summable bounded Fredholm modules. Using the unbounded Kasparov product, we
exhibit a family of generalized spectral triples, possessing mildly unbounded
commutators, whilst still giving well defined -homology classes.Comment: 67 pages, minor changes in Section 5.1 and 6.
Toeplitz operators on -spaces of a tree
Let be a rooted, countable infinite tree without terminal vertices. In
the present paper, we characterize the spectra, self-adjointness and positivity
of Toeplitz operators on the spaces of -summable functions on . Moreover,
we obtain a necessary and sufficient condition for Toeplitz operators to have
finite rank on such function spaces
The differentiation operator on discrete function spaces of a tree
In this paper, we study the differentiation operator acting on discrete
function spaces; that is spaces of functions defined on an infinite rooted
tree. We discuss, through its connection with composition operators, the
boundedness and compactness of this operator. In addition, we discuss the
operator norm and spectrum, and consider when such an operator can be an
isometry. We then apply these results to the operator acting on the discrete
Lipschitz space and weighted Banach spaces, as well as the Hardy spaces defined
on homogeneous trees
Dirac operators and spectral triples for some fractal sets built on curves
We construct spectral triples and, in particular, Dirac operators, for the
algebra of continuous functions on certain compact metric spaces. The triples
are countable sums of triples where each summand is based on a curve in the
space. Several fractals, like a finitely summable infinite tree and the
Sierpinski gasket, fit naturally within our framework. In these cases, we show
that our spectral triples do describe the geodesic distance and the Minkowski
dimension as well as, more generally, the complex fractal dimensions of the
space. Furthermore, in the case of the Sierpinski gasket, the associated
Dixmier-type trace coincides with the normalized Hausdorff measure of dimension
.Comment: 48 pages, 4 figures. Elementary proofs omitted. To appear in Adv.
Mat
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