Commutativity of the Valdivia–Vogt table of representations of function spaces

In this article, we show that the Valdivia–Vogt structure table—containing the sequence space representations of the most used spaces of smooth functions appearing in the theory of distributions—can be interpreted as a commutative diagram, i.e., there is an isomorphism between the space of infinitely differentiable functions and the space , where s is the space of rapidly decreasing sequences, such that its restriction to the other function spaces in the structure table yields an isomorphism between these spaces of smooth functions and their sequence space representation. This result answers the corresponding question of Prof. Dietmar Vogt formulated on the conference “Functional Analysis: Applications to Complex Analysis and Partial Differential Equations” held in B?dlewo in May 2012.     

Integral operators commuting with dilations and rotations in generalized Morrey-type spaces

We find conditions for the boundedness of integral operators $K$ commuting with dilations and rotations in a local generalized Morrey space. We also show that under the same conditions, these operators preserve the subspace of such Morrey space, known as vanishing Morrey space. We also give necessary conditions for the boundedness when the kernel is non-negative. In the case of classical Morrey spaces, the obtained sufficient and necessary conditions coincide with each other. In the one-dimensional case, we also obtain similar results for global Morrey spaces. In the case of radial kernels, we also obtain stronger estimates of $Kf$ via spherical means of $f$. We demonstrate the efficiency of the obtained conditions for a variety of examples such as weighted Hardy operators, weighted Hilbert operator, their multidimensional versions, and others.     

Surjectivity of differential operators and the division problem in certain function and distribution spaces

We give an overview on surjectivity conditions for partial differential operators and operators defined by multiplication with polynomials on certain function and distribution spaces of Laurent Schwartz. We complement the classical results by treating the surjectivity of operators on the space of slowly increasing functions and on the space of rapidly decreasing distributions, respectively.     

Absolute continuity of convolutions of orbital measures on Riemannian symmetric spaces

We study the absolute continuity of the measures and of on the Riemannian symmetric spaces X of noncompact type for nonzero elements Xj, X∈a. For m,l?r+1, where r is the rank of X, the considered convolutions have a density. We conjecture that the condition m,l?r+1 is necessary. The conjecture is proved for the symmetric spaces of type An−1. Moreover, the minimal value of l is determined, in function of the irregularity of X.     

Weakly prime sets for function spaces

We define and study weakly prime sets for a function space and show that it coincides with the known concept of weakly prime sets for function algebras and spaces of affine functions.     

Fixed and periodic point theory in certain spaces

This paper proves the existence of periodic and fixed points for contractive conditions in metric (symmetric) space.     

Extremely non-complex spaces

We show that there exist infinite-dimensional extremely non-complex Banach spaces, i.e. spaces X such that the norm equality Id+T²=1+T² holds for every bounded linear operator . This answers in the positive Question 4.11 of [V. Kadets, M. Martín, J. Merí, Norm equalities for operators on Banach spaces, Indiana Univ. Math. J. 56 (2007) 2385–2411]. More concretely, we show that this is the case of some C(K) spaces with few operators constructed in [P. Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004) 151–183] and [G. Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004) 217–239]. We also construct compact spaces K₁ and K₂ such that C(K₁) and C(K₂) are extremely non-complex, C(K₁) contains a complemented copy of C(2^ω) and C(K₂) contains a (1-complemented) isometric copy of ℓ_∞.     

Compressions of resolvents and maximal radius of regularity

Suppose that is left invertible in for all , where is an open subset of the complex plane. Then an operator-valued function is a left resolvent of in if and only if has an extension , the resolvent of which is a dilation of of a particular form. Generalized resolvents exist on every open set , with included in the regular domain of . This implies a formula for the maximal radius of regularity of in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by
J. Zemánek is obtained.

     

Fixed point theorems in symmetric spaces and applications to probabilistic spaces

Several fixed point theorems for symmetric spaces are proved. Then these theorems are used in symmetric PPM-space to prove and generalize Theorem 6 of [T.L. Hicks, B.E. Rhoades, Fixed point theory in symmetric spaces with applications to probabilistic spaces, Nonlinear Anal. 36 (1999) 331-344].     

Uniqueness of symmetric basis in quasi-Banach spaces

We show that if X is a nonlocally convex natural quasi-Banach space with symmetric basis whose Banach envelope is isomorphic to ?₁, then all symmetric bases of X are equivalent. The scope of this result is quite ample since the Banach envelopes of natural quasi-Banach spaces with basis always exhibit an ?₁-like behavior, in the sense that they contain copies of 's uniformly complemented.     

Complexifications of symmetric spaces and Jordan theory

Generalizing Hermitian and pseudo-Hermitian spaces, we define twisted complex symmetric spaces, and we show that they correspond to an algebraic object called Hermitian Jordan triple products. The main topic of this work is to investigate the class of real forms of twisted complex symmetric spaces, called the category of symmetric spaces with twist. We show that this category is equivalent to the category of all real Jordan triple systems, and we can use a work of B.O. Makarevic in order to classify the irreducible spaces. The classification shows that most irreducible symmetric spaces have exactly one twisted complexification. This leads to open problems concerning the relation of Jordan and Lie triple systems.

     

Sampling and interpolation in large Bergman and Fock spaces

We obtain sampling and interpolation theorems in weighted spaces of analytic functions for radial weights of arbitrary (more rapid than polynomial) growth. We give an application to invariant subspaces of arbitrary index in large weighted Bergman spaces.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

Sampling numbers and function spaces

We want to recover a continuous function using only its function values. Let us assume, that f is from the unit ball of some function space (for example a fractional Sobolev space or a Besov space) and the precision of the reconstruction is measured in the norm of another function space of this type. We describe the rate of convergence of the optimal sampling method (linear as well as nonlinear) in this setting.     

Hypercyclic convolution operators on Fréchet spaces of analytic functions

A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on Cn, which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the Schatten-von Neumann class.     

Geometric flow and rigidity on symmetric spaces of noncompact type

In this paper we show that, under a suitable condition, every nonsingular geometric flow on a manifold which is modeled on the Furstenberg boundary of , where is a symmetric space of non-compact type, induces a torus action, and, in particular, if the manifold is a rational homology sphere, then the flow has a closed orbit.

     

Products of weighted composition and differentiation operators into weighted Zygmund and Bloch spaces

We characterize boundedness and compactness of products of differentiation operators and weighted composition operators between weighted Banach spaces of analytic functions and weighted Zygmund spaces or weighted Bloch spaces with general weights.     

Interpolation orbits in couples of Lebesgue spaces

The paper deals with interpolation orbits for linear operators acting from a arbitrary couple { (U ₀), (U ₁)} of weighted L _p spaces into an arbitrary couple { (V ₀), (V ₁)} of such spaces, where 1 p ₀,p ₁,q ₀,q ₁ . Here L _p (U) is the space of measurable functions f on a measure space such that fU L _p , equipped with the norm . The paper describes the orbits of arbitrary elements a (U ₀) + (U ₁). It contains proofs of the results announced in C. R. Acad. Sci. Paris, Ser. I, 334, 881–884 (2002).__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 56–68, 2005Original Russian Text Copyright © by V. I. OvchinnikovTranslated by V. I. Ovchinnikov