Weak eigenvectors and the functional calculus

The operator Lry = / + Py, whose domain is determined in part by the Stieltjes integral boundary condition Jo dv{i)y{f) = 0, is studied in Xj¡($>, 1), 1 < p < oo.It is shown that Lp has a dense domain; hence there exists a dual operator L* operating on .£¡¡(0,1).After finding LJ we show that both L, and L¡¡ are Fredholm operators.This implies some elementary results concerning the spectrum and states of Lp.Finally two eigenfunction expansions are derived.

The equations governing a three-layer beam in which the middle layer is viscoelastic were derived by means of a variational principle. These equations were solved subject to arbitrary time dependent boundary conditions of both the simply-supported and cantilever type. In the simply-supported case, an eigenfunction expansion was constructed after the governing equations had been transformed through operation by a differential operator. In the cantilever case, the solution was obtained by Fourier transform with respect to time, and inversion by residue theory. An analysis of the theoretical response of the cantilever beam to steady state sinusoidal loading was carried out, and an experiment was conducted for comparison.

This paper is devoted to the multivariable \(H^\infty \) functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if \((A_1,\ldots , A_d)\) is such a family, if \(A_k\) is R-sectorial of R-type \(\omega _k\in (0,\pi )\), \(k=1,\ldots ,d\), and if \((A_1,\ldots , A_d)\) admits a bounded \(H^\infty (\Sigma _{\theta _1}\times \cdots \times \Sigma _{\theta _d})\) joint functional calculus for some \(\theta _k\in (\omega _k,\pi )\), then it admits a bounded \(H^\infty (\Sigma _{\theta _1}\times \cdots \times \Sigma _{\theta _d})\) joint functional calculus for all \(\theta _k\in (\omega _k,\pi )\), \(k=1,\ldots ,d\). Second we introduce square functions adapted to the multivariable case and extend to this setting some of the well-known one-variable results relating the boundedness of \(H^\infty \) functional calculus to square function estimates. Third, on K-convex reflexive spaces, we establish sharp dilation properties for d-tuples \((A_1,\ldots , A_d)\) which admit a bounded \(H^\infty (\Sigma _{\theta _1}\times \cdots \times \Sigma _{\theta _d})\) joint functional calculus for some \(\theta _k<\frac{\pi }{2}\).

On suitable Banach spaces, the existence of a certain functional calculus for an operator is equivalent to that operator having a certain integral repreaentation with respect to some family of projections. In this paper we consider the problem of identifying the precise classes of spaces on which these equivalences hold, for several of the most important functional calculi.

In this chapter, we explain (1) oscillation of solutions of a homogeneous second-order linear differential equation (§VI-1), (2) the Sturm-Liouville problems (§§VI2—VI-4, topics including Green’s functions, self-adjointness, distribution of eigen-values, and eigenfunction expansion), (3) scattering problems (§§VI-5—VI-9, mostly focusing on reflectionless potentials), and (4) periodic potentials (§VI-10). The materials concerning these topics are also found in [CL, Chapters 7, 8, and 11], [Har2 Chapter XI], [Copl, Chapter 1], [Be12, Chapter 6], and [TD]. Singular self-adjoint boundary-value problems (in particular continuous spectrum, limit-point and limit-circle cases) are not explained in this book. For these topics, see [CL, Chapter 9].

In this paper we shall construct a certain spectral theoryfor closed linear operators on a Banach space.These operators have a suitable spectral behaviour on subsets of theirspectra but we must eliminate some residual part which do not offer informationabout the intimate structure of the considered objects, at least from our pointof view.It will be easy to see that this theory contains many examples of operators,bounded or not, having a functional calculus on their spectrum [1], [2], [3], [6],[8], [9].A permanent model for our construction will be the theory of decomposableoperators on a Banach space [7], [2].Throughout this paper the sets of points will be taken in CΌo^Cuf

Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems, enjoy a number of properties which make them a natural analog of the classical orthogonal polynomials. In the present paper we continue their study. The most important new properties are their hypergeometric representations which allow us to derive their generating functions and in some cases also Mehler-Heine type formulas.

Introduction. This article considers a fairly general class of operators on sections of a vector bundle over a compact manifold, including the differential operators and singular integral operators. The members of this class share many of the properties of differential operators, particularly the elliptic ones. Two general advantages have motivated the development. First, it leads to transparent proofs of the familiar results for elliptic equations, on regularity, the Fredholm alternative, and eigenfunction expansions; and for a larger class than the differential operators. These proofs are not new; rather some of the techniques used in the case of differential operators appear here as general properties of the class of integro-differential operators considered. A second advantage of the larger system, not extensively exploited in this article, is topological. Homotopies (in the class of smooth functions) of the characteristic polynomial of a differential operator can be lifted to homotopies of the operator itself in the class of integro-differential operators considered, but not (generally) in the class of differential operators. This is an important help in treating some questions raised by Gelfand [6]; some of the questions concerning the index have now been answered by Atiyah and Singer [1]. To find the notation and main results, one can read ?1-?3 (except for proofs), ?6, and the definitions and statements of theorems and corollaries from the remaining sections. The paper is organized as follows. ?1 describes the well-known function spaces on Rn that are involved, as well as certain operators on them. ?2 describes the singular integral operators and their symbols. ?3 extends this collection to one that contains the differential operators on R, as well as the inverses of the invertible elliptic operators. The symbol u(A) of an operator A is defined, and the behavior of a under composition of operators is discussed. ?4 considers the behavior of a under coordinate changes. ?5 gives some necessary lemmas from functional analysis. ?6 establishes the notation for vector bundles, and the analogs for bundles of the function spaces of ?1. ?7 defines the singular integral operators on sections of a vector bundle E over a compact manifold X, and their symbols. If A is a singular integral operator from sections of one bundle E

We study the fractional differential equation (*) Dαu(t) + BDβu(t) + Au(t) = f(t), 0 ⩽ t ⩽ 2π (0 ⩽ β < α ⩽ 2) in periodic Lebesgue spaces Lp(0, 2π; X) where X is a Banach space. Using functional calculus and operator valued Fourier multiplier theorems, we characterize, in UMD spaces, the well posedness of (*) in terms of R-boundedness of the sets {(ik)α((ik)α + (ik)βB + A)−1}k∈Z and {(ik)βB((ik)α + (ik)βB + A)−1}k∈Z. Applications to the fractional problems with periodic boundary condition, which includes the time diffusion and fractional wave equations, as well as an abstract version of the Basset-Boussinesq-Oseen equation are treated. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The present paper is concerned with the essential spectrum of the singular matrix differential operator of mixed order over the interval (a, b), where D = d/dt. It is shown that if the coefficients are locally integrable functions and T is in the limit circle case, then the essential spectrum of T is given by the essential range of ${{p(t)d(t)-\vert h(t)\vert^2}\over{p(t)}} $\end{document}. The main idea is to transform the original spectral problem into a spectral problem of a singular Hamiltonian differential operator so that the classical Titchmarsh‐Weyl theory can apply. The approach used here can be applied to many other cases. © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim

Essential spectra of singular matrix differential operators of mixed order

L. Waelbroeck gives a holomorphic functional calculus for Banach algebras and analytic functions on Banach spaces. The properties of this calculus extend the well-known properties for the case of several complex variables. In this last situation, W. Zame has obtained a theorem of unicity where the famous condition of compatibility is dropped. We obtain a theorem analogous to Zame’s for Waelbroeck’s calculus restricted to a certain algebra of germs of functions. We consider Banach spaces whose topological duals have the bounded approximation property. Also, results of the same kind as above are given for bornological algebras.

This chapter deals with strongly continuous n-parameter groups W of bounded operators in a Banach space F. The principal topics are certain scales of Banach spaces and the functional calculus that can be associated to such groups.KeywordsHilbert SpaceBanach SpaceBesov SpaceFunctional CalculusInterpolation SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let –iA be the generator of a C 0-group \({(U(s))_{s\in \mathbb {R}}}\) on a Banach space X and ω > θ(U), the group type of U. We prove a transference principle that allows to estimate \({||{f(A)}||}\) in terms of the \({{\rm{L}}^{p}(\mathbb {R};X)}\)-Fourier multiplier norm of \({f(\cdot \pm i \omega)}\). If X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev–de Laubenfels. If X is a UMD space, one obtains a bounded \({{\rm{H}}^\infty_1}\)-calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded \({{\rm{H}}^\infty}\)-calculus on sectors.

On integral formulas and their applications in functional calculus