Invariant subspaces and extremum problems in spaces of vector-valued functions

Shift-type invariant subspaces of contractions

This paper is devoted to some Korovkin approximation results in cones of Hausdorff continuous set-valued functions and in spaces of vector valued functions. Some classical results are exposed in order to give a more complete treatment of the subject. New contributions are concerned both with the general theory than in particular with the so-called convexity monotone operators, which are considered in cones of set-valued function and also in spaces of vector-valued functions.

We analyse the strong connections between spaces of vector-valued Lipschitz functions and spaces of linear continuous operators. We apply these links to study duality, Schur properties and norm attainment in the former class of spaces as well as in their canonical preduals.

We discuss new results concerning unbounded Toeplitz operators defined in Segal-Bargmann spaces of (vector-valued) functions, i.e. the space of all entire functions which are square summable with respect to the Gaussian measure in $\mathrm{C}^n$. The problem of finding adjoints of analytic Toeplitz operators is solved in some cases. Closedness of the range of analytic Toeplitz operators is studied. We indicate an example of an entire function inducing a Toeplitz operator, for which the space of polynomials is not a core though it is contained in its domain.

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.

We introduce a vector-valued version of a uniform algebra, called the vector-valued function space over a uniform algebra. The diameter two properties of the vector-valued function space over a uniform algebra on an infinite compact Hausdorff space are investigated. Every nonempty relatively weakly open subset of the unit ball of a vector-valued function space $$A(K, (X, \tau ))$$ over an infinite dimensional uniform algebra has diameter two, where $$\tau $$ is a locally convex Hausdorff topology on a Banach space X compatible to a dual pair. Under the assumption of X equipped with the norm topology being uniformly convex and the additional condition that $$A\otimes X\subset A(K, X)$$ , it is shown that Daugavet points and $$\Delta $$ -points on A(K, X) over a uniform algebra A are the same, and they are characterized by the norm-attainment at a limit point of the Shilov boundary of A. In addition, a sufficient condition for the convex diametral local diameter two property of A(K, X) is also provided. Similar results also hold for an infinite dimensional uniform algebra.

A pair of functions defined on a set X with values in a vector space E is said to be disjoint if at least one of the functions takes the value 0 at every point in X. An operator acting between vector-valued function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. We characterize compact and weakly compact disjointness preserving operators between spaces of Banach space-valued differentiable functions.

In [1] it was proved that the unit ball of a reflexive Banach space has an uncountable set of extreme points. In this note it is shown that this set is also massive in some topological sense.

The instability index of a linear operator with point spectrum is defined to be the total multiplicity of its eigenvalues with positive real parts. Under certain conditions the computation of the instability index of an unbounded nonsymmetric operator acting in a Hilbert space can be reduced to an analogous problem for a certain selfadjoint operator. It is shown that if the original operator is a differential operator acting in a space of vector-valued functions with a single variable, then the selfadjoint operator corresponding to it can be found in the form of an integro-differential operator which can be realized constructively by solving a special elliptic boundary value problem. Analogues of known theorems of Morse on the connection between the instability index and the number of conjugate points are established for this integro-differential operator.Bibliography: 26 titles.

We classify extreme points and exposed points of the unit ball of the space of bilinear symmetric forms on the real Banach space of bilinear symmetric forms on $l_{\infty}^2.$ It is shown that for this case, the set of extreme points is equal to the set of exposed points.

We investigate pointwise multipliers on vector-valued function spaces over \({\mathbb {R}}^d\), equipped with Muckenhoupt weights. The main result is that in the natural parameter range, the characteristic function of the half-space is a pointwise multiplier on Bessel-potential spaces with values in a UMD Banach space. This is proved for a class of power weights, including the unweighted case, and extends the classical result of Shamir and Strichartz. The multiplication estimate is based on the paraproduct technique and a randomized Littlewood–Paley decomposition. An analogous result is obtained for Besov and Triebel–Lizorkin spaces.

In an earlier paper [1] the writer showed the development of a variational method for starlike functions and applied this method to a general class of extremal problems connected with the coefficients of the function. It is the purpose of this paper to study the classes of extremal problems to which this variational method can be applied and to make some remarks as to its utility. The class S* of starlike functions is the class of all functions analytic and schlicht in the unit circle, normalized by f(0) = 0 and f'(0) =1, and mapping the unit circle onto a domain which is starshaped with respect to the origin. In [1] it was shown that given any function f(z) S* a variation can be made within the class S* giving a change in the function

In this paper, we study the well-posedness of the third-order differential equation with finite delay (P3): αu‴ (t) + u″(t) = Au(t) + Bu′ (t) + Fut + f(t)(t ∈ $$\mathbb{T}$$ := [0, 2π]) with periodic boundary conditions u(0) = u(2π), u′(0) = u′(2π), u″ (0) = u″(2π), in periodic Lebesgue–Bochner spaces Lp( $$\mathbb{T}$$ ;X) and periodic Besov spaces Bp,qs( $$\mathbb{T}$$ ;X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B) ≠{0}, α ≠ 0 is a fixed constant and F is a bounded linear operator from Lp([−2π, 0];X) (resp. Bp,qs([−2π, 0];X)) into X, ut is given by ut(s) = u(t + s) when s ∈ [−2π, 0]. Necessary and sufficient conditions for the Lp-well-posedness (resp. Lp-well-posedness) of (P3) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.

A series of results similar to Marcinkiewicz's theorem on the interpolation of operators is put forward. The difference from the classical forms of this theorem is that spaces of integrable functions are replaced by some function classes that are extensions of various Hardy spaces. Some applications of these results to the extension of Carleson's embedding theorem and the Hardy-Littlewood inequalities for analytic functions in Hardy classes are presented. Bibliography: 41 titles.

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.