On the Oka principle in a Banach space,II

 Let X be one of the Banach spaces c ₀ , ℓ _p , 1≤p<∞; Ω⊂X pseudoconvex open, a holomorphic Banach vector bundle with a Banach Lie group G ^* for structure group. We show that a suitable Runge-type approximation hypothesis on X, G ^* (which we also prove for G ^* a solvable Lie group) implies the vanishing of the sheaf cohomology groups H ^q (Ω, 𝒪 ^E ), q≥1, with coefficients in the sheaf of germs of holomorphic sections of E. Further, letting 𝒪^Γ (𝒞^Γ) be the sheaf of germs of holomorphic (continuous) sections of a Banach Lie group bundle Γ→Ω with Banach Lie groups G, G ^* for fiber group and structure group, we show that a suitable Runge-type approximation hypothesis on X, G, G ^* (which we prove again for G, G ^* solvable Lie groups) implies the injectivity (and for X=ℓ₁ also the surjectivity) of the Grauert–Oka map H ¹ (Ω, 𝒪^Γ)→H ¹ (Ω, 𝒞^Γ) of multiplicative cohomology sets. Received: 1 March 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 32L20, 32L05, 46G20 RID="*" ID="*" Kedves Laci Móhan kisfiamnak. RID="*" ID="*" To my dear little Son     

A converse to the Oka principle over certain complex Banach manifolds

First Kajiwara then Leiterer gave geometric or cohomological criteria in the spirit of the Grauert-Oka principle for an open subset D of a Stein manifold M to be itself Stein. We give here criteria analogous to Leiterer's, e.g., for a relatively open subset D of a closed complex Hilbert submanifold M of separable Hilbert space to be itself biholomorphic to a closed complex Hilbert submanifold of separable Hilbert space.     

On Runge approximation in a Banach space

We show that on some open sets, more general than balls, Runge approximation is possible in certain Banach spaces, and also in certain complex Banach manifolds. We also show that there is an entire holomorphic curve in Hilbert space on which there is a bounded holomorphic function on the trace of a ball that has no bounded holomorphic extension to even a smaller concentric ball. Using the same technique we also prove that a form of Runge approximation better than an error function is not always possible.     

On normal crossings in a Banach space

We show that if A is a closed complex analytic subset of a Banach space X with an unconditional basis (e.g., X=?₂) that has only normal crossings for singularities, then the structure and ideal sheaves of A are cohesive sheaves over X, and consequently, they are acyclic over any pseudoconvex open subset of X.     

On Bessel systems in a Banach space

We consider Bessel systems in a Banach space and study their operator and projective characteristics. Using an operator construction for the continuation of the system, we prove an analog of Schur’s theorem on continuable systems.     

On the stabilizability problem in Banach space

The model is a linear system defined on Banach (state and control) spaces, with the operator acting on the state only the infinitesimal generator of a strongly continuous semigroup. The stabilizability problem of expressing the control through a bounded operator acting on the state as to make the resulting feedback system globally asymptotically stable is considered. On the negative side, and in contrast with the finite dimensional theory, a few counter examples are given of systems which are densely controllable in the space and yet are not stabilizable, even if some further “nice properties” hold. Use is made of the notion of essential spectrum and its stability under relatively compact perturbations. On the positive side, it is shown, however, that for large classes of systems of physical interest (classical selfadjoint boundary value problems, delay equations, etc.) controllability on a suitable finite dimensional subspace still yields stabilizability on the whole space.     

An existence principle for a periodic solution of a differential equation in Banach space

The equation d²x/dt²=Ax +f(t, x) is considered in a Banach space E, where A is a fixed unbounded linear operator, andf(t, x) is a nonlinear operator which is periodic in t and satisfies a Lipschitz condition with respect to x E. Existence conditions have been obtained for a well defined generalized periodic solution of this equation, and also when this solution coincides with the true solution. Similar results have been obtained for the first order equation.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 105–112, July, 1968.     

Extremal solutions and comparison principle for nonlinear integro-differential equations in a Banach space

In this paper, we consider an initial value problem for nonlinear integro-differential equations in a Banach space. First, we give a comparison result between the under and over functions and some comparison principles. Then, using these results and the Kuratowski measure of noncompactness, we establish the existence theorem of extremal solutions between the under and over functions, and prove that there exists a unique solution between the lower and upper solutions under an additional Lipschitz's condition. Project supported by the Natural Science Foundation of Shandong Province.     

The invariance principle for Banach space valued random variables

We extend the invariance principle to triangular arrays of Banach space valued random variables, and as an application derive the invariance principle for lattices of random variables. We also point out how the q-dimensional time parameter Yeh-Wiener process is naturally related to a one dimensional time Wiener process with an infinite dimensional Banach space as a state space.     

An extension to Banach space of Pontryagin's maximum principle

An extension of Pontryagin's maximum principle to the case where the state space is infinite dimensional is given. The control process is governed by ordinary nonlinear differential equations. A property of control processes, which is analogous to well-known, nonlinear interior mapping theorems, makes up the basis for the proofs.     

On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements

The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and C^∗-algebra elements.     

On a nonlinear Volterra integral equation in a Banach space

The equation u(t) + ∝₀^tk(t ? s)g(s) ds?f(t), t ? 0, is studied in a real Banach space with uniformly convex dual. Conditions, sufficient for the existence of a unique solution, are given for the operatorvalued kernel k, the nonlinear m-accretive operators g(t) and the function f. The case when k is realvalued, g(t) ≡ g and X a reflexive Banach space is also considered. These results extend earlier results by Barbu, Londen and MacCamy.     

On a Banach space approximable by Jacobi polynomials

Let X represent either the space C[-1,1] L _p ^(α,β) (w), 1 ≦ p < ∞ on [-1, 1]. Then Xare Banach spaces under the sup or the p norms, respectively. We prove that there exists a normalized Banach subspace X ¹ _αβ of Xsuch that every f ∈ X ¹ _αβ can be represented by a linear combination of Jacobi polynomials to any degree of accuracy. Our method to prove such an approximation problem is Fourier–Jacobi analysis based on the convergence of Fourier–Jacobi expansions. This revised version was published online in June 2006 with corrections to the Cover Date.