We study the problem of how a map f:M → ℝ on an n-manifold M induces canonically an affinor $A\left(f\right):T{T}^{\left(r\right)}M\to T{T}^{\left(r\right)}M$ on the vector r-tangent bundle $T^{\left(r\right)}M=\left(J^r(M,ℝ)₀\right)*$ over M. This problem is reflected in the concept of natural operators $A:T_{|ℳfₙ}^{(0,0)}⇝T^{(1,1)}{T}^{\left(r\right)}$. For integers r ≥ 1 and n ≥ 2 we prove that the space of all such operators is a free (r+1)²-dimensional module over ${}^{\infty }\left(T^{\left(r\right)}ℝ\right)$ and we construct explicitly a basis of this module.

Let n,r,k be natural numbers such that n ≥ k+1. Non-existence of natural operators $C^r₀⟿Q\left(reg{T}_k^r\to K_k^r\right)$ and $C^r₀⟿Q\left(reg{T}_k^{r*}\to K_k^{r*}\right)$ over n-manifolds is proved. Some generalizations are obtained.

All natural operators A transforming a linear vector field X on a vector bundle E into a vector field A(X) on the r-jet prolongation $J^{r}E$ of E are given. Similar results are deduced for the r-jet prolongations $J_v^{r}E$ and $J^{\left[r\right]}E$ in place of $J^{r}E$.

Let F:ℳ f→ ℬ be a vector bundle functor. First we classify all natural operators $T_{|ℳfₙ}⇝T^{(0,0)}\left(F_{|ℳfₙ}\right)*$ transforming vector fields to functions on the dual bundle functor $\left(F_{|ℳfₙ}\right)*$. Next, we study the natural operators $T{*}_{|ℳfₙ}⇝T*\left(F_{|ℳfₙ}\right)*$ lifting 1-forms to $\left(F_{|ℳfₙ}\right)*$. As an application we classify the natural operators $T{*}_{|ℳfₙ}⇝T*\left(F_{|ℳfₙ}\right)*$ for some well known vector bundle functors F.

Results of Emmanuele and Drewnowski are used to study the containment of c₀ in the space $K_{w*}(X*,Y)$, as well as the complementation of the space $K_{w*}(X*,Y)$ of w*-w compact operators in the space $L_{w*}(X*,Y)$ of w*-w operators from X* to Y.

We prove that the problem of finding all $ℳf_m$-natural operators $B:Q⟿Q{T}^A$ lifting classical linear connections ∇ on m-manifolds M to classical linear connections $B_M\left(\nabla \right)$ on the Weil bundle $T^{A}M$ corresponding to a p-dimensional (over ℝ) Weil algebra A is equivalent to the one of finding all $ℳf_m$-natural operators $C:Q⟿(T{¹}_{p-1},T*\otimes T*\otimes T)$ transforming classical linear connections ∇ on m-manifolds M into base-preserving fibred maps $C_M\left(\nabla \right):T{¹}_{p-1}M={⨁}_M^{p-1}TM\to T*M\otimes T*M\otimes TM$.

Let r and n be natural numbers. For n ≥ 2 all natural operators $T_{|ℳfₙ}⇝T*T^{r*}$ transforming vector fields on n-manifolds M to 1-forms on $T^{r*}M=J^r(M,ℝ)₀$ are classified. For n ≥ 3 all natural operators $T_{|ℳfₙ}⇝\Lambda ²T*T^{r*}$ transforming vector fields on n-manifolds M to 2-forms on $T^{r*}M$ are completely described.

We show that every operator from ${\ell }_s$ to ${\ell }_p\otimes ̂{\ell }_q$ is compact when 1 ≤ p,q < s and that every operator from ${\ell }_s$ to ${\ell }_p\widehat{\otimes ̂}{\ell }_q$ is compact when 1/p + 1/q > 1 + 1/s.

Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators ${}^{b,c}$ is bounded on the Dirichlet spaces ${}_{p,a}$. We also give a short and direct proof of boundedness of ${}^{b,c}$ on the Hardy space $H^p$ for 1 < p < ∞.

We classify, up to isomorphism, the spaces of compact operators (E,F), where E and F are the Banach spaces of all continuous functions defined on the compact spaces $2^{}\times [0,\alpha ]$, the topological products of Cantor cubes $2^{}$ and intervals of ordinal numbers [0,α].

Some properties and applications of natural vector bundle morphisms $T_A\circ {⨂}_s^q\to {⨂}_s^q\circ T_A$ over $i{d}_{T_A}$ are presented.

A classification of weakly compact multiplication operators on $L\left(L_p\right),$1<p<$,isgiven.ThisanswersaquestionraisedbySaksmanandTylliin1992.Theclassificationinvolvestheconceptof$p$-strictlysingularoperators,andwealsoinvestigatethestructureofgeneral$p$-strictlysingularoperatorson$Lp$.Themainresultisthatifanoperator$T$on$Lp$,$1<p<2$,is$p$-strictlysingularand$T|X$isanisomorphismforsomesubspace$X$of$Lp$,then$X$embedsinto$Lr$forall$r<2$,but$X$neednotbeisomorphictoaHilbertspace.$It is also shown that if $T$ is convolution by a biased coin on $L_p$ of the Cantor group, $1\le p<2$, and $T_{|X}$ is an isomorphism for some reflexive subspace $X$ of $L_p$, then $X$ is isomorphic to a Hilbert space. The case $p=1$ answers a question asked by Rosenthal in 1976.

We describe all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝Q{T}^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D\left(\nabla \right)$ on cotangent bundles $T^{*}Y$ of $Y$. We show that this problem can be reduced to finding ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝(T^{*},{\otimes }^p{T}^{*}\otimes {\otimes }^{q}T)$ for $p=2$, $q=1$ and $p=3$, $q=0$.

We classify all $ℱ{ℳ}_{m,n}$-natural operators $:J²⟿J²V^A$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $\left(\Gamma \right):V^{A}Y\to J²V^{A}Y$ on the vertical Weil bundle $V^{A}Y\to M$ corresponding to a Weil algebra A.

Let $ℳf_m$ be the category of $m$-dimensional manifolds and local diffeomorphisms and  let $T$ be the tangent functor on $ℳf_m$. Let $𝒱$ be the category of real vector spaces and linear maps and let ${𝒱}_m$ be the category of $m$-dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors $F:{𝒱}_m\to 𝒱$ admitting $ℳf_m$-natural operators $\tilde{J}$ transforming classical linear connections $\nabla$ on $m$-dimensional manifolds $M$ into almost complex structures $\tilde{J}\left(\nabla \right)$ on $F\left(T\right)M={\bigcup }_{x\in M}F\left(T_{x}M\right)$.

Let m and r be natural numbers and let $P^r:ℳf_m\to ℱℳ$ be the rth order frame bundle functor. Let $F:ℳf_m\to ℱℳ$ and $G:ℳf_k\to ℱℳ$ be natural bundles, where $k=dim\left(P^r{ℝ}^m\right)$. We describe all $ℳf_m$-natural operators A transforming sections σ of $FM\to M$ and classical linear connections ∇ on M into sections A(σ,∇) of $G\left(P^{r}M\right)\to P^{r}M$. We apply this general classification result to many important natural bundles F and G and obtain many particular classifications.

We introduce and study a new concept of strongly $l_p$-summing m-linear operators in the category of operator spaces. We give some characterizations of this notion such as the Pietsch domination theorem and we show that an m-linear operator is strongly $l_p$-summing if and only if its adjoint is $l_p$-summing.

Let $M$ be an $m$-dimensional manifold and $A={𝔻}_k^r/I=ℝ\oplus N_A$ a Weil algebra of height $r$. We prove that any $A$-covelocity $T_x^{A}f\in T_x^{A*}M$, $x\in M$ is determined by its values over arbitrary $max\{\mathrm{width}A,m\}$ regular and under the first jet projection linearly independent elements of $T_x^{A}M$. Further, we prove the rigidity of the so-called universally reparametrizable Weil algebras. Applying essentially those partial results we give the proof of the general rigidity result $T^{A*}M\simeq T^{r*}M$ without coordinate computations, which improves and generalizes the partial...

Let 1 ≤ p < ∞, $={\left(Xₙ\right)}_{n\in \mathbb{N}}$ be a sequence of Banach spaces and $l_p\left(\right)$ the coresponding vector valued sequence space. Let $={\left(Xₙ\right)}_{n\in \mathbb{N}}$, $={\left(Yₙ\right)}_{n\in \mathbb{N}}$ be two sequences of Banach spaces, $={\left(Vₙ\right)}_{n\in \mathbb{N}}$, Vₙ: Xₙ → Yₙ, a sequence of bounded linear operators and 1 ≤ p,q < ∞. We define the multiplication operator $M_{}:l_p\left(\right)\to l_q\left(\right)$ by $M_{}\left({\left(xₙ\right)}_{n\in \mathbb{N}}\right):={\left(Vₙ\left(xₙ\right)\right)}_{n\in \mathbb{N}}$. We give necessary and sufficient conditions for $M_{}$ to be 2-summing when (p,q) is one of the couples (1,2), (2,1), (2,2), (1,1), (p,1), (p,2), (2,p), (1,p), (p,q); in the last case 1 < p < 2, 1 < q < ∞. ...

In this paper, we obtain new sufficient conditions for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ to be univalent in the open unit disc $𝒰$, where the functions $f_1,f_2,...,f_n$ belong to the classes $S^{*}(a,b)$ and $𝒦(a,b)$. The order of convexity for the operators $F_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ and $G_{{\alpha }_1,{\alpha }_2,...,{\alpha }_n,\beta }\left(z\right)$ is also determined. Furthermore, and for $\beta =1$, we obtain sufficient conditions for the operators $F_n\left(z\right)$ and $G_n\left(z\right)$ to be in the class $𝒦(a,b)$. Several corollaries and consequences of the main results are also considered.

Let $ℐ$ be a coherent subsheaf of a locally free sheaf $𝒪\left(E_0\right)$ and suppose that $ℱ=𝒪\left(E_0\right)/ℐ$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $ℱ$ we construct a vector-valued Coleff-Herrera current $\mu$ with support on the variety associated to $ℱ$ such that $\phi$ is in $ℐ$ if and only if $\mu \phi =0$. Such a current $\mu$ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....

We classify all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $A$ transforming projectable-projectable torsion-free classical linear connections $\nabla$ on fibered-fibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$ into $r$th order Lagrangians $A\left(r\right)$ on the fibered-fibered linear frame bundle $L^{fib-fib}\left(Y\right)$ on $Y$. Moreover, we classify all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $B$ transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds $Y$ of dimension $(m_1,m_2,n_1,n_2)$ into Euler morphism $B\left(\nabla \right)$ on $L^{fib-fib}\left(Y\right)$. These classifications can be expanded on...

We give a classification of all linear natural operators transforming $p$-vectors (i.e., skew-symmetric tensor fields of type $(p,0)$) on $n$-dimensional manifolds $M$ to tensor fields of type $(q,0)$ on $T^{A}M$, where $T^A$ is a Weil bundle, under the condition that $p\ge 1$, $n\ge p$ and $n\ge q$. The main result of the paper states that, roughly speaking, each linear natural operator lifting $p$-vectors to tensor fields of type $(q,0)$ on $T^A$ is a sum of operators obtained by permuting the indices of the tensor products of linear natural...

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant...