14 research outputs found
A note on the spectrum of Lipschitz operators and composition operators on Lipschitz spaces
Fix a metric space and let be the Banach space of
complex-valued Lipschitz functions defined on . A weighted composition
operator on is an operator of the kind , where and are any map.
When such an operator is bounded, it is actually the adjoint operator of a
so-called weighted Lipschitz operator acting on the
Lipschitz-free space . In this note, we study the spectrum of
such operators, with a special emphasize when they are compact. Notably, we
obtain a precise description in the non-weighted case: the
spectrum is finite and made of roots of unity
Peller's problem concerning Koplienko-Neidhardt trace formulae: the unitary case
We prove the existence of a complex valued -function on the unit circle,
a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class
, such that the perturbated operator does not belong to the
space of trace class operators. This resolves a problem of Peller
concerning the validity of the Koplienko-Neidhardt trace formula for unitaries
Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae
A formula for the norm of a bilinear Schur multiplier acting from the
Cartesian product of two copies of the
Hilbert-Schmidt classes into the trace class is established in
terms of linear Schur multipliers acting on the space of
all compact operators. Using this formula, we resolve Peller's problem on
Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice
continuously differentiable function with a bounded second derivative, a
self-adjoint (unbounded) operator and a self-adjoint operator such that
f(A+B)-f(A)-\frac{d}{dt}(f(A+tB))\big\vert_{t=0}\notin \mathcal S^1. $
Compact and weakly compact Lipschitz operators
International audienceAny Lipschitz map between two pointed metric spaces may be extended in a unique way to a bounded linear operator between their corresponding Lipschitz-free spaces. In this paper, we give a necessary and sufficient condition for f to be compact in terms of metric conditions on . This extends a result by A. Jiménez-Vargas and M. Villegas-Vallecillos in the case of non-separable and unbounded metric spaces. After studying the behavior of weakly convergent sequences made of finitely supported elements in Lipschitz-free spaces, we also deduce that f is compact if and only if it is weakly compact
A note on the spectrum of Lipschitz operators and composition operators on Lipschitz spaces
Fix a metric space M and let Lip 0 (M) be the Banach space of complex-valued Lipschitz functions defined on M. A weighted composition operator on Lip 0 (M) is an operator of the kind wC f : g → w • g • f , where w : M → C and f : M → M are any map. When such an operator is bounded, it is actually the adjoint operator of a so-called weighted Lipschitz operator w f acting on the Lipschitz-free space F (M). In this note, we study the spectrum of such operators, with a special emphasize when they are compact. Notably, we obtain a precise description in the non-weighted w ≡ 1 case: the spectrum is finite and made of roots of unity
A PRE-ADJOINT APPROACH ON WEIGHTED COMPOSITION OPERATORS BETWEEN SPACES OF LIPSCHITZ FUNCTIONS
We consider weighted composition operators, that is operators of the type , acting on spaces of Lipschitz functions. Bounded weighted composition operators, as well as some compact weighted composition operators, have been characterized quite recently. In this paper, we provide a different approach involving their pre-adjoint operators, namely the weighted Lipschitz operators acting on Lipschitz free spaces. This angle allows us to improve some results from the literature. Notably, we obtain a distinct characterization of boundedness with a precise estimate of the norm. We also characterise injectivity, surjectivity, compactness and weak compactness in full generality