THE STRONG IRREDUCIBILITY OF A CLASS OF COWEN–DOUGLAS OPERATORS ON BANACH SPACES

Let ${\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ be the set of Cowen–Douglas operators of index $n$ on a nonempty bounded connected open subset $\unicode[STIX]{x1D6FA}$ of $\mathbb{C}$. We consider the strong irreducibility of a class of Cowen–Douglas operators ${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ on Banach spaces. We show ${\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})\subseteq {\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ and give some conditions under which an operator $T\in {\mathcal{F}}{\mathcal{B}}_{n}(\unicode[STIX]{x1D6FA})$ is strongly irreducible. All these results generalise similar results on Hilbert spaces.

CESÀRO–VOLTERRA PATH INTEGRAL FORMULA ON A SURFACE

If a symmetric matrix field e = (eij) of order three satisfies the Saint-Venant compatibility relations in a simply-connected open subset Ω of ℝ3, then e is the linearized strain tensor field of a displacement field v of Ω, i.e. [Formula: see text] in Ω. A classical result, due to Cesàro and Volterra, asserts that, if the field e is smooth, the unknown displacement field v(x) at any point x ∈ Ω can be explicitly written as a path integral inside Ω with endpoint x, and whose integrand is an explicit function of the functions eij and their derivatives. Now let ω be a simply-connected open subset in ℝ2 and let θ : ω → ℝ3 be a smooth immersion. If two symmetric matrix fields (γαβ) and (ραβ) of order two satisfy appropriate compatibility relations in ω, then (γαβ) and (ραβ) are the linearized change of metric and change of curvature tensor field corresponding to a displacement vector field η of the surface θ(ω). We show here that a "Cesàro–Volterra path integral formula on a surface" likewise holds when the fields (γαβ) and (ραβ) are smooth. This means that the displacement vector η(y) at any point θ(y), y ∈ ω, of the surface θ(ω) can be explicitly computed as a path integral inside ω with endpoint y, and whose integrand is an explicit function of the functions γαβ and ραβ and their covariant derivatives. Such a formula has potential applications to the mathematical analysis and numerical simulation of linear "intrinsic" shell models.

C∞-REGULARITY OF A MANIFOLD AS A FUNCTION OF ITS METRIC TENSOR

A basic theorem from differential geometry asserts that if the Riemann curvature tensor associated with a smooth field C of positive-definite symmetric matrices of order n vanishes in a simply-connected open subset Ω of ℝn, then C is the metric tensor of a manifold isometrically immersed in ℝn. If Ω is connected, then the isometric immersion Θ defined in this fashion is unique up to isometries of ℝn. We prove that if the set Ω is bounded and has a smooth boundary, then the mapping C ↦ Θ is of class C∞ between manifolds in appropriate Banach spaces.

RECOVERY OF A SURFACE WITH BOUNDARY AND ITS CONTINUITY AS A FUNCTION OF ITS TWO FUNDAMENTAL FORMS

If a field A of class [Formula: see text] of positive-definite symmetric matrices of order two and a field B of class [Formula: see text] of symmetric matrices of order two satisfy together the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of ℝ2, then there exists an immersion [Formula: see text], uniquely determined up to proper isometries in ℝ3, such that A and B are the first and second fundamental forms of the surface θ(ω). Let [Formula: see text] denote the equivalence class of θ modulo proper isometries in ℝ3 and let [Formula: see text] denote the mapping determined in this fashion. The first objective of this paper is to show that, if ω satisfies a certain "geodesic property" (in effect a mild regularity assumption on the boundary of ω) and if the fields A and B and their partial derivatives of order ≤ 2 (respectively, ≤ 1), have continuous extensions to [Formula: see text], the extension of the field A remaining positive-definite on [Formula: see text], then the immersion θ and its partial derivatives of order ≤ 3 also have continuous extensions to [Formula: see text]. The second objective is to show that, if ω satisfies the geodesic property and is bounded, the mapping ℱ can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces [Formula: see text] for the continuous extensions of the matrix fields (A, B), and [Formula: see text] for the continuous extensions of the immersions θ.

Similarity Classification of Cowen-Douglas Operators

Let ℋ be a complex separable Hilbert space and ℒ(ℋ) denote the collection of bounded linear operators on ℋ. An operator A in ℒ(ℋ) is said to be strongly irreducible, if , the commutant of A, has no non-trivial idempotent. An operator A in ℒ(ℋ) is said to be a Cowen-Douglas operator, if there exists Ω, a connected open subset of C, and n, a positive integer, such that(a)Ω ⊂ σ(A) = ﹛z ∈ C | A – z not invertible﹜;(b)ran(A – z) = ℋ, for z in Ω;(c)Vz∈Ω ker(A – z) = ℋ and(d)dim ker(A – z) = n for z in Ω.In the paper, we give a similarity classification of strongly irreducible Cowen-Douglas operators by using the K0-group of the commutant algebra as an invariant.

PLANAR LACES

Let S be a connected open subset of 2-sphere S2 which is identified with the extended plane R2 ∪ {∞}. We assume that S contains the n segments {1, 2, …, n} × [-1, 1]. An n-lace ℓ (in S) is the union ℓ1 ∪ … ∪ ℓn of disjoint simple arcs in S such that ∂ ℓi = {(i, 1), (π (i),-1)}, i = 1, …, n, for some permutation π of {1, 2, …, n}. In this paper we will do present mapping class group description of n-laces, and 1-lace in 1-punctured plane, and 2-laces in S2. We will also describe the isotropy subgroup of the trivial lace in [Formula: see text].

THE OSCILLATION PATTERN OF SOLUTIONS TO PARABOLIC EQUATIONS AS TIME GOES TO INFINITY

We consider the semilinear parabolic equation [Formula: see text] where Ω be a bounded, connected open subset of ℝN with a Lipschitz continuous boundary and f is a locally Lipschitz continuous function: ℝ→ℝ. If u is a bounded solution de (1.1) for which the ω-limit set satisfies [Formula: see text] the number of connected components of {x∈Ω; φ(x)≠=0}) is equal to a constant n∞ on ω(u) and there exists T>0 such that for all t≥T, the set {x∈Ω; u(t,x)≠0} has a finite number of connected components precisely equal to n∞.

Separators in Continuous Images of Ordered Continua and Hereditarily Locally Connected Continua

Let X be a Hausdorff space which is the continuous image of an ordered continuum. We prove that every irreducible separator of X is metrizable. This is a far reaching extension of the 1967 theorem of S. Mardešić which asserts that X has a basis of open sets with metrizable boundaries. Our first result is then used to show that, in particular, if Y is an hereditarily locally connected continuum, then for subsets of Y quasi-components coincide with components, and that the boundary of each connected open subset of Y is accessible by ordered continua. These results answer open problems in the literature due to the fourth and third authors, respectively.

The Proximal Subgradient and Constancy

If f is a lower semicontinuous function mapping a connected open subset of ℝn to (—∞, ∞], and if the proximal subgradient of f reduces to zero wherever it exists, then f is constant.