The paper defines and studies the Drazin inverse for a closed linear operator in a Banach space in the case that belongs to a spectral set of the spectrum of . Results are applied to extend a result of Krein on a nonhomogeneous second order differential equation in a Banach space.
To any bounded analytic semigroup on Hilbert space or on -space, one may associate natural ’square functions’. In this survey paper, we review old and recent results on these square functions, as well as some extensions to various classes of Banach spaces, including noncommutative -spaces, Banach lattices, and their subspaces. We give some applications to functional calculus, similarity problems, multiplier theory, and control theory.
For an operator in a possibly infinite-dimensional Hilbert space of a certain class, we set down axioms of an abstract intersection theory, from which the Riemann hypothesis regarding the spectrum of that operator follows. In our previous paper (2011) we constructed a GNS (Gelfand-Naimark-Segal) model of abstract intersection theory. In this paper we propose another model, which we call a standard model of abstract intersection theory. We show that there is a standard model of abstract intersection...
We give a necessary and a sufficient condition for a subset of a locally convex Waelbroeck algebra to have a non-void left joint spectrum In particular, for a Lie subalgebra we have if and only if generates in a proper left ideal. We also obtain a version of the spectral mapping formula for a modified left joint spectrum. Analogous theorems for the right joint spectrum and the Harte spectrum are also valid.
We present a new method to study the classical fractional integrals of Weyl. This new approach basically consists in considering these operators in the largest space where they make sense. In particular, we construct a theory of fractional integrals of Weyl by studying these operators in an appropriate Fréchet space. This is a function space which contains the -spaces, and it appears in a natural way if we wish to identify these fractional operators with fractional powers of a suitable non-negative...
Let Y be a Banach space and let be a subspace of an space, for some p ∈ (1,∞). We consider two operators B and C acting on S and Y respectively and satisfying the so-called maximal regularity property. Let ℬ and be their natural extensions to . We investigate conditions that imply that ℬ + is closed and has the maximal regularity property. Extending theorems of Lamberton and Weis, we show in particular that this holds if Y is a UMD Banach lattice and is a positive contraction on for any...
A study is made of a symmetric functional calculus for a system of bounded linear operators acting on a Banach space. Finite real linear combinations of the operators have real spectra, but the operators do not necessarily commute with each other. Analytic functions of the operators are formed by using functions taking their values in a Clifford algebra.
Kato’s conjecture, stating that the domain of the square root of any accretive operator with bounded measurable coefficients in is the Sobolev space , i.e. the domain of the underlying sesquilinear form, has recently been obtained by Auscher, Hofmann, Lacey, McIntosh and the author. These notes present the result and explain the strategy of proof.