Paley–Littlewood decomposition for sectorial operators and interpolation spaces

We prove Paley–Littlewood decompositions for the scales of fractional powers of 0‐sectorial operators A on a Banach space which correspond to Triebel–Lizorkin spaces and the scale of Besov spaces if A is the classical Laplace operator on We use the ‐calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace‐type operators on manifolds and graphs, Schrödinger operators and Hermite expansion. We also give variants of these results for bisectorial operators and for generators of groups with a bounded ‐calculus on strips.     

An abstract transmission problem in a thin layer, I: Sharp estimates

We study an elliptic transmission problem in Banach spaces. The problem is considered on the juxtaposition of two intervals, one of which of small length δ, and models physical phenomena in media constituted by two parts with different physical characteristics. We obtain results of existence, uniqueness, maximal regularity and optimal dependence on the parameter δ for Lp solutions of the problem. The main tools of our approach are impedance and admittance operators (i.e. Dirichlet-to-Neumann and Neumann-to-Dirichlet operators) and H^∞ functional calculus for sectorial operators in Banach spaces.     

Operators with an absolute functional calculus

We study sectorial operators with a special type of functional calculus, which we term an absolute functional calculus. A typical example of such an operator is an invertible operator A (defined on a Banach space X) considered on the real interpolation space (Dom(A), X) _θ,p where 0 < θ < 1 and 1 < p < ∞. In general the absolute functional calculus can be characterized in terms of real interpolation spaces. We show that operators of this type have a strong form of the H ^∞-calculus and behave very well with respect to the joint functional calculus. We give applications of these results to recent work of Arendt, Batty and Bu on the existence of Hölder-continuous solutions for the abstract Cauchy problem.     

Solvability of Elliptic Differential Equations,Set in Three Habitats with Skewness Boundary Conditions at the Interfaces

In this work, we study an elliptic differential equation set in three habitats with skewness boundary conditions at the interfaces. It represents the linear stationary case of dispersal problems of population dynamics which incorporate responses at interfaces between the habitats. Existence, uniqueness and regularity of the solution of these problems are obtained in Hölder spaces under necessary and sufficient conditions on the data. Our techniques are based on the semigroup theory, the fractional powers of linear operators, the \(H^{\infty }\) functional calculus for sectorial operators in Banach spaces and some properties of real interpolation spaces.     

Area Integral Functions and H Functional Calculus for Sectorial Operators on Hilbert Spaces

Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi's results on H^∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.     

On fixed point theorems of mixed monotone operators and applications

In this paper, we obtain some new existence and uniqueness theorems of positive fixed point of mixed monotone operators in Banach spaces partially ordered by a cone. Finally we apply the main theorem to a nonlinear parabolic partial differential equation. Some results are new even for increasing or decreasing operators.     

The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space

We consider linear fractional differential operator equations involving the Caputo derivative. The goal of this paper is to establish conditions for the unique solvability of the inverse Cauchy problem for these equations. We use properties of the Mittag-Leffler function and the calculus of sectorial operators in a Banach space. For equations with operators in a general form we obtain sufficient conditions for the unique solvability, and for equations with densely defined sectorial operators we obtain necessary and sufficient unique solvability conditions.     

r-ASYMPTOTICALLY QUASI-FINITE RANK OPERATORS AND THE SPECTRUM OF MEASURES

Abstract

The r-asymptotically quasi finite rank operators were introduced in [10]. For regular operators on Banach lattices, these operators are the order theoretic analogue of Riesz operators on Banach spaces. We establish their basic properties and apply these in the spectral analysis of convolution operators.     

A Joint Functional Calculus for Sectorial Operators with Commuting Resolvents

In this paper we study the notion of joint functional calculusassociated with a couple of resolvent commuting sectorial operatorson a Banach space X. We present some positive results when Xis, for example, a Banach lattice or a quotient of subspacesof a B-convex Banach lattice. Furthermore, we develop a notionof a generalized H-functional calculus associated with the extensionto (H) of a sectorial operator on a B-convex Banach lattice, where H is a Hilbert space. We apply our results to a newconstruction of operators with a bounded H-functional calculusand to the maximal regularity problem. 1991 Mathematics SubjectClassification: 47A60, 47D06, 46C15.     

Existence theorems for monotone operators in reflexive Banach spaces

In this paper, we obtain an existence theorem for single-valued monotone operators in a reflexive Banach space. Using this result, we prove a fixed point theorem for nonexpansive mappings in a Hilbert space and an existence theorem for maximal monotone operators in a Banach space. Received: 3 July 2006 Revised: 15 January 2007     

Functional calculus on real interpolation spaces for generators of C0‐groups

We study functional calculus properties of C₀‐groups on real interpolation spaces using transference principles. We obtain interpolation versions of the classical transference principle for bounded groups and of a recent transference principle for unbounded groups. Then we show that each group generator on a Banach space has a bounded ‐calculus on real interpolation spaces. Additional results are derived from this.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

A functional calculus for analytic generators ofC 0-groups

In [9] and [3] anF(S )-functional calculus for sectorial operators is constructed via the Dunford-Riesz integral. This calculus implicitely defines the well-known complex powers of such operators. Sectorial operators with bounded imaginary powers turn out to be of particular interest due to the remarkable Dore-Venni theorem. In [12] this theorem is proved via the theory of analytic generators ofC ₀-groups. These results suggest the existence ofF(S )-functional, calculi forC ₀-groups and their analytic generators. In this paper we show that such functional calculi indeed exsist, however the approach via the Dunford-Riesz integral is no longer viable. Therefore a different approach via an approximation argument is introduced. Existence and uniqueness theorems are given and we show how the functional calculi relate to known results. Examples illustrate the theory.     

Feynman’s operational calculi for noncommuting operators: The monogenic calculus

In recent papers the authors presented their approach to Feynman’s operational calculi for a system of not necessarily commuting bounded linear operators acting on a Banach space. The central objects of the theory are the disentangling algebra, a commutative Banach algebra, and the disentangling map which carries this commutative structure into the noncommutative algebra of operators. Under assumptions concerning the growth of disentangled exponential expressions, the associated functional calculus for the system of operators is a distribution with compact support which we view as the joint spectrum of the operators with respect to the disentangling map. In this paper, the functional calculus is represented in terms of a higher-dimensional analogue of the Riesz-Dunford calculus using Clifford analysis.     

Singular Integral Operators with Fixed Singularities on Weighted Lebesgue Spaces

The paper is devoted to study of singular integral operators with fixed singularities at endpoints of contours on weighted Lebesgue spaces with general Muckenhoupt weights. Compactness of certain integral operators with fixed singularities is established. The membership of singular integral operators with fixed singularities to Banach algebras of singular integral operators on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights is proved on the basis of Balakrishnans formula from the theory of strongly continuous semi-groups of closed linear operators. Symbol calculus for such operators, Fredholm criteria and index formulas are obtained.     

The functional calculus of full operators with discrete spectrum

We construct the functional calculus for full operators with discrete spectrum over Banach spaces in the interpolation classes of symbols associated with given operators. We describe new classes of full operators in Banach spaces. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998 pp. 127–135.     

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

Area integral functions for sectorial operators on L spaces

Area integral functions are introduced for sectorial operators on L^p-spaces. We establish the equivalence between the square and area integral functions for sectorial operators on L^p spaces. This follows that the results of Cowling, Doust, McIntosh, Yagi, and Le Merdy on H^infin functional calculus of sectorial operators on L^p-spaces hold true when the square functions are replaced by the area integral functions.