Finite-dimensional Banach spaces with numerical index zero.
Finite-rank perturbations of positive operators and isometries
We completely characterize the ranks of A - B and for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that of . For...
First results on spectrally bounded operators
A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is defined to be spectrally bounded if there is a constant M ≥ 0 such that r(Tx) ≤ Mr(x) for all x ∈ E, where r(·) denotes the spectral radius. We study some basic properties of this class of operators, which are sometimes analogous to, sometimes very different from, those of bounded operators between Banach spaces.
Fixed point theorems based on Leray-Schauder degree
Fixed points of holomorphic mappings for domains in Banach spaces.
Floquet operators without absolutely continuous spectrum
Foliations transverse to fibers of Seifert manifolds.
Fonction spectrale d'opérateurs non elliptiques
Fonctions entières et théorèmes de Banach-Steinhaus dans certaines algèbres complètes
Form perturbations of the second quantized Dirac field.
Formeln für Pole der Resolvente und den Radius eines wesentlichen Spektrums.
Forms, functional calculus, cosine functions and perturbation
In this article we describe properties of unbounded operators related to evolutionary problems. It is a survey article which also contains several new results. For instance we give a characterization of cosine functions in terms of mild well-posedness of the Cauchy problem of order 2, and we show that the property of having a bounded -calculus is stable under rank-1 perturbations whereas the property of being associated with a closed form and the property of generating a cosine function are not....
Formulae for joint spectral radii of sets of operators
The formula is proved for precompact sets M of weakly compact operators on a Banach space. Here ϱ(M) is the joint spectral radius (the Rota-Strang radius), is the Hausdorff spectral radius (connected with the Hausdorff measure of noncompactness) and r(M) is the Berger-Wang radius.
Four characterizations of scalar-type operators with spectrum in a half-line
-scalar-type spectrality criterions for operators A whose resolvent set contains the negative reals are provided. The criterions are given in terms of growth conditions on the resolvent of A and the semigroup generated by A. These criterions characterize scalar-type operators on the Banach space X if and only if X has no subspace isomorphic to the space of complex null-sequences.
Fourier-like methods for equations with separable variables
It is well known that a power of a right invertible operator is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However, a linear combination of right invertible operators (in particular, their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The...
Fractional Laplacian with singular drift
For α ∈ (1,2) we consider the equation , where b is a time-independent, divergence-free singular vector field of the Morrey class . We show that if the Morrey norm is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.
Fractional Powers of Almost Non-Negative Operators
Mathematics Subject Classification: Primary 47A60, 47D06.In this paper, we extend the theory of complex powers of operators to a class of operators in Banach spaces whose spectrum lies in C ]−∞, 0[ and whose resolvent satisfies an estimate ||(λ + A)(−1)|| ≤ (λ(−1) + λm) M for all λ > 0 and for some constants M > 0 and m ∈ R. This class of operators strictly contains the class of the non negative operators and the one of operators with polynomially bounded resolvent. We also prove that this theory...
Fractional powers of non-densely defined operators
Fractional powers of non-negative operators in Fréchet spaces.
Fredholm determinants
The article provides with a down to earth exposition of the Fredholm theory with applications to Brownian motion and KdV equation.