C0-Groups and Functional Calculi

In this paper, we aim to present an advantage of using K-normed space (or called cone normed space) for investigating Cauchy problem in a scale of Banach spaces {(F s, ||.||s) : s ∈ (0, 1]}.

In this paper, by developing a new method we advance to prove the existence results for the Cauchy problem with delay $$d_tu=f(t,u(t),u(h(t))),\;t\in (0,T),\;u(0)=u_0$$, in a scale of Banach spaces. The arguments are based upon Darbo–Sadovskii fixed point theorem for condensing operators on Frechet spaces and the technique of vector-valued measure of noncompactness.

In the second chapter we will study well-posedness of the linear, time-dependent Cauchy problem $$ \frac{{du}}{{dt}}(t) = A(t)u(t),u({t_0}) = {u_0}$$ for operators A(t) ∈ ∩k∈IN 0 ℒ(Xk+m, Xk) of order m in a scale of Banach spaces (Xk) k (i.e., Xk for k ∈ ℕ0 is a Banach space with Xk ↪ Xl for k ≥ l). Here well-posedness roughly means that for sufficiently smooth initial values u0 there are unique solutions depending continuously on u0.

In this paper we are concerned with the problem of constructing the functional calculus for an operator with continuous spectrum from a given collection of weak eigenvectors.A functional calculus is a homomorphism/- Tf of an algebra of functions defined over the 'spectrum' V into a subalgebra of bounded linear operators acting in a banach space E; we may regard an 'eigenfunction expansion' for such a functional calculus as an expression for all Tf over a banach space D dense in E of the form Tf= vf(X)Txdm(X), where m is a measure over Fand Tx is a bounded linear operator from D into the space of 'distributions' D'.In general, it is hoped that the range of Tx, lmTx, will consist of rather precisely specified weak eigenvectors (specified, for example, by boundary values).Since a family of weak eigenvectors is often available, a problem that often appears in applications is the following: Given a family of weak eigenvectors at X, when is it true that this family coincides with lmTxl There are many results in the literature for a closed operator with dense domain in a banach space with continuous spectrum which guarantee a rich functional calculus and a representation of the functional calculus in terms of an eigenfunction expansion where, in fact, ImTA consists of weak eigenvectors.F. Browder [3] established such results for a large class of nonsymmetric partial differential operators, and Gel'fand and Silov [5] were the first to consider the problem of the existence of eigenfunction expansions in general operator theoretic terms.However there are few results that tell us which weak eigenvectors to choose for the eigenfunction expansion.Most results of this nature are confined to differential operators with strong regularity theorems, the most classical being Weyl's construction of the Plancherel formula for a singular second order ordinary differential operator [10].The crucial step is the 'Weyl lemma' which asserts that certain weak eigenvectors are actually infinitely differentiable, and so, classical uniqueness theorems may be invoked to determine precisely which weak eigenvectors contribute to the eigenfunction expansion.The results given here, in contrast, are well within the context of functional

Peano type theorem for abstract parabolic equations

Solvability of differential equations in scales of Banach spaces

The dynamics of an infinite system of point particles in $$\mathbb {R}^d$$ , which hop and interact with each other, is described at both micro- and mesoscopic levels. The states of the system are probability measures on the space of configurations of particles. For a bounded time interval $$[0,T)$$ , the evolution of states $$\mu _0 \mapsto \mu _t$$ is shown to hold in a space of sub-Poissonian measures. This result is obtained by: (a) solving equations for correlation functions, which yields the evolution $$k_0 \mapsto k_t$$ , $$t\in [0,T)$$ , in a scale of Banach spaces; (b) proving that each $$k_t$$ is a correlation function for a unique measure $$\mu _t$$ . The mesoscopic theory is based on a Vlasov-type scaling, that yields a mean-field-like approximate description in terms of the particles’ density which obeys a kinetic equation. The latter equation is rigorously derived from that for the correlation functions by the scaling procedure. We prove that the kinetic equation has a unique solution $$\varrho _t$$ , $$t\in [0,+\infty )$$ .

We consider parabolic problems with non-Lipschitz nonlinearities in different scales of Banach spaces and prove local-in-time existence theorems. A new class of parabolic equations that have analytic solutions is obtained.

Non-autonomous rough semilinear PDEs and the multiplicative Sewing lemma

Chapter 4. Nonlinear Operator Volterra Equations in the Scales of Banach Spaces

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 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

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.

The purpose of this work is to study various geometric properties of Banach spaces, with particular emphasis on their stability with respect to the formation of (infinite) absolute sums and Köthe-Bochner spaces of vector- valued functions. The first three chapters are devoted to the class of acs (``alternatively convex or smooth'') Banach spaces and their relatives, the luacs and uacs spaces. These notions were introduced by Kadets et al. in 2000 (in connection with the so called anti-Daugavet property) and form common generalisations of the usual convexity and smoothness properties of Banach spaces. In the first chapter, these notions (as well as some other related properties introduced by the author) are studied from a general point of view. For example, some equivalent characterisations are established and results on duality and quotient spaces are discussed. In the second chapter, various results on (mainly infinite) absolute sums of acs spaces and their relatives are proved, while in the third chapter, acs properties of Köthe-Bochner spaces of vector-valued functions are discussed. The fourth chapter is devoted to the so called Opial property and its variants, as well as some other geometric notions (WORTH property and Garcia-Falset coefficient) of Banach spaces, which are all related to the fixed point property for nonexpansive mappings. Some stability results for these properties in certain infinite absolute sums are established. Furthemore, some properties analogous to the Opial property in Lebesgue-Bochner spaces and Ces`aro spaces of vector-valued functions are discussed. The fifth chapter concerns Banach spaces with the so called ``ball generated property'' (BGP). The stability of this property with respect to certain sums of two spaces is proved. The last chapter deals with generalised lush (GL) spaces, which were introduced by Huang et al. in 2013 in connection with the Mazur-Ulam extension problem. Some results on GL-spaces are established. In particular, every M-ideal in a GL-space is again a GL-space.