Core–EP inverse

In this work, we introduce a notion of ‘core–EP inverse’ for a square matrix which is not essentially of index one. This extends the notion of ‘core inverse’, which was initially defined for the matrices of index one. The properties of matrices having ‘core–EP inverse’ and ‘core–EP generalized inverse’ are studied, and obtained a formula to compute the core–EP generalized inverse from a particular linear combination of minors of given matrix.     

The existence of an inverse limit of an inverse system of measure spaces — A purely measurable case

The existence of an inverse limit of an inverse system of (probability) measure spaces has been investigated since the very beginning of the modern probability theory. Results from Kolmogorov [11], Bochner [2], Choksi [6], Metivier [15], Bourbaki [4], Mallory and Sion [12] among others have paved the way of the deep understanding of this problem. All the above results, however, call for some topological concepts, or at least the ones which are closely related topological ones. In this paper we investigate purely measurable inverse systems of (probability) measure spaces, and give a sufficient condition for the existence of a unique inverse limit. An example for the considered purely measurable inverse systems of (probability) measure spaces is also given.     

Weighted Fourier–Laplace transforms in reproducing kernel Hilbert spaces on the sphere

We study the action of a weighted Fourier–Laplace transform on the functions in the reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel on the sphere. After defining a notion of smoothness implied by the transform, we show that smoothness of the kernel implies the same smoothness for the generating elements (spherical harmonics) in the Mercer expansion of the kernel. We prove a reproducing property for the weighted Fourier–Laplace transform of the functions in the RKHS and embed the RKHS into spaces of smooth functions. Some relevant properties of the embedding are considered, including compactness and boundedness. The approach taken in the paper includes two important notions of differentiability characterized by weighted Fourier–Laplace transforms: fractional derivatives and Laplace–Beltrami derivatives.     

Core–EP pre-order of Hilbert space operators

A new binary relation associated with the core–EP inverse is presented and studied on the corresponding subset of all generalized Drazin invertible bounded linear Hilbert space operators. Using the (dual) core partial order between core parts of operators and the minus partial order between quasinilpotent parts of operators, new pre-orders and partial orders are also introduced and characterized.     

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator spaces.     

Estimates of singular numbers of a Hilbert–Schmidt integral operator

We consider the estimates for singular numbers of a Hilbert?CSchmidt integral operator in terms of the continuity moduli of its kernel.     

Weighted variational inequalities in non-pivot Hilbert spaces with applications

We introduce variational inequalities defined in non-pivot Hilbert spaces and we show some existence results. Then, we prove regularity results for weighted variational inequalities in non-pivot Hilbert space. These results have been applied to the weighted traffic equilibrium problem. The continuity of the traffic equilibrium solution allows us to present a numerical method to solve the weighted variational inequality that expresses the problem. In particular, we extend the Solodov-Svaiter algorithm to the variational inequalities defined in finite-dimensional non-pivot Hilbert spaces. Then, by means of a interpolation, we construct the solution of the weighted variational inequality defined in a infinite-dimensional space. Moreover, we present a convergence analysis of the method.     

Consequences of the existence of a non-compact operator between nuclear Köthe spaces

It is proved that if a continuous, linear, non-compact operator TnK(A)K(B) exists between two nuclear Köthe spaces then there exists a common (up to isomorphism) step space of the two spaces provided K(A) is regular and K(B) is isomorphic to a subspace of K(A) or K(B) is regular and K(A) is isomorphic to a quotient space of K(B) or K(A) and K(B) satisfy a certain splitting condition. Consequences in some particular cases are also obtained.     

Commutators of fractional maximal operator on generalized Orlicz–Morrey spaces

In the present paper, we shall give necessary and sufficient conditions for the Spanne and Adams type boundedness of the commutators of fractional maximal operator on generalized Orlicz–Morrey spaces, respectively. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.     

Sequences of Weighted Composition Operators on the Hardy–Hilbert Space

Let \(\varphi \) be an analytic self map of the open unit disc \(\mathbb {D}\). Assume that \(\psi \) is an analytic map of \(\mathbb {D}\). Suppose that f is in the Hardy–Hilbert space of the open unit disc \(H^2\). The operator that takes f into \(\psi \cdot f \circ \varphi \) is a weighted composition operator, and is denoted by \(C_{\psi ,\varphi }\). In this paper we relate the convergence of the sequence \(\{ C_{\psi _n,\varphi _n}\}\) in different operator topologies to the convergence of the two sequences of maps \(\{\varphi _n \}\) and \(\{ \psi _n\}\).     

ℂ-differentiability of mappings of Hilbert spaces

The process of differentiation is studied for a class of mappings of regions of a Hilbert space. We introduce the concept of Cl-differentiability of a mapping at a point and establish sufficient conditions of Cl-differentiability.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 3, pp. 424–428, March, 1995.     

Moore–Penrose inverse of an invertible infinite matrix

In this article we show that, contrary to finite matrices (with real or complex entries) an invertible infinite matrix V could have a Moore–Penrose inverse that is not a classical inverse of V. This also answers a recent open problem on infinite matrices.     

Imbeddings between Weighted Orlicz–Lorentz Spaces

We establish necessary and sufficient conditions for imbeddings of weighted Orlicz–Lorentz spaces.     

Weighted composition operators between μ-Bloch spaces on the unit ball

In this paper,necessary and sufficient conditions are given for the weighted composition operator T_(ψ,ψ) to be bounded or compact from the space β_μto β_ν(or β_(μ,0) to β_(ν,0) ) on the unit ball of C~n.At the same time,a series of corollaries are also obtained.     

Weighted composition operators between μ -bloch spaces on the unit ball

In this paper, necessary and sufficient conditions are given for the weighted composition operator T _ψ,φ to be bounded or compact from the space β_μ to β_υ (or β_μ,0 to β _υ,0)on the unit ball of C ^rn. At the same time, a series of corollaries are also obtained.     

q–differ-integral operator on p–valent functions associated with operator on Hilbert space

Making use of multivalent functions with negative coefficients of the type f (z)=z^p-^∑∞_k=p+1a_kz^k,which are analytic in the open unit disk and applying the q-derivative a q–differintegral operator is considered.Furthermore by using the familiar Riesz-Dunford integral,a linear operator on Hilbert space H is introduced.A new subclass of p-valent functions related to an operator on H is defined.Coefficient estimate,distortion bound and extreme po...