L-weakly and M-weakly compact operators and the centre

We extend known results concerning the centre of spaces of regular (resp. weakly compact or compact) operators between two Banach lattices to the setting of L-weakly compact and M-weakly compact operators. We also show that the L-weakly compact, M-weakly compact, and compact operators lying in the centre of a Banach lattice coincide.     

Triangularizability of Polynomially Compact Operators

An operator on a complex Banach space is polynomially compact if a non-zero polynomial of the operator is compact, and power compact if a power of the operator is compact. Theorems on triangularizability of algebras (resp. semigroups) of compact operators are shown to be valid also for algebras (resp. semigroups) of polynomially (resp. power) compact operators, provided that pairs of operators have compact commutators.     

Some Results on b-AM-Compact Operators

We characterize Banach lattices for which the class of order weakly compact operators (resp. b-weakly compact operators) between Banach lattices coincide with that of b-AM-compact operators.     

The duality problem for the Class of b-weakly compact operators

We establish necessary and sufficient conditions under which b-weakly compact operators between Banach lattices have b-weakly compact adjoints or operators with b-weakly compact adjoints are themselves b-weakly compact. Also, we give some consequences.      

Invariant Subspaces of Operator Lie Algebras and the Theory of K-Algebras

It is proved that if a Lie algebra of compact operators contains a nonzero ideal consisting of quasinilpotent operators then this Lie algebra has a nontrivial invariant subspace. Some applications of this result to lattices of invariant subspaces for families of compact operators and to structures of ideals of Banach Lie algebras with compact adjoint action are given.     

Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action

It is proved that a Lie algebra of compact operators with a non-zero Volterra ideal is reducible (has a nontrivial invariant subspace). A number of other criteria of reducibility for collections of operators is obtained. The results are applied to the structure theory of Lie algebras of compact operators and normed Lie algebras with compact adjoint action.     

Banach格上的b-几乎Dunford-Pettis算子

首先给出了Banah格上的b-几乎Dunford-Pettis算子的定义;其次,研究了b-几乎Dunford-Pettis算子的相关性质,如b-几乎Dunford-Pettis算子的等价刻画,构成空间的性质,以及控制性;最后,研究了b-几乎DunfordPettis算子与相关算子(b-弱紧算子,弱紧算子,几乎Dunford-Pettis算子)间的关系.     

On the weak compactness of b-weakly compact operators

We characterize Banach lattices on which the class of b-weakly compact operators coincides with that of weakly compact operators.     

Weakly compact operators on non-complete normed spaces

We prove that weakly compact operators on a non-reflexive normed space cannot be bijective. We also show that, in the above result, bijectivity cannot be relaxed to surjectivity. Finally, we study the behaviour of surjective weakly compact operators on a non-reflexive normed space, when they are perturbed by small scalar multiples of the identity, and derive from this study the recent result of Spurný [A note on compact operators on normed linear spaces, Expo. Math. 25 (2007) 261–263] that compact operators on an infinite-dimensional normed space cannot be surjective.     

The b-Weak Compactness of Order Weakly Compact Operators

We characterize Banach lattices on which the class of order weakly compact operators coincides with that of b-weakly compact operators.     

U-ideals of factorable operators

We suggest a method of renorming of spaces of operators which are suitably approximable by sequences of operators from a given class. Further we generalize J. Johnsons's construction of ideals of compact operators in the space of bounded operators and observe e.g. that under our renormings compact operators are u-ideals in the: space of 2-absolutely summing operators or in the space of operators factorable through a Hilbert space.     

Some Properties of the Class of b-Weakly Compact Operators

We establish several characterizations of b-weakly compact operators one of them is similar to that established by Meyer-Nieberg for order weakly compact operators.     

On compact extensions of multilinear operators

In this paper we prove that, under certain conditions, Nicodemi extensions of compact multilinear operators between Banach spaces are compact as well. An application of this result to the isometric/isomorphic theory of spaces of compact multilinear operators is provided.     

Lipschitz compact operators

We introduce the notion of Lipschitz compact (weakly compact, finite-rank, approximable) operators from a pointed metric space X into a Banach space E. We prove that every strongly Lipschitz p-nuclear operator is Lipschitz compact and every strongly Lipschitz p-integral operator is Lipschitz weakly compact. A theory of Lipschitz compact (weakly compact, finite-rank) operators which closely parallels the theory for linear operators is developed. In terms of the Lipschitz transpose map of a Lipschitz operator, we state Lipschitz versions of Schauder type theorems on the (weak) compactness of the adjoint of a (weakly) compact linear operator.     

Fredholm Operators and Riesz Theory for Polynomially Compact Operators

In this article, we give a characterization of a class of bounded Fredholm operators on a Banach space which is developed to present some general existence results of the operators equation of the second kind. The obtained results are used to describe the Riesz–Schauder theory of compact operators in the more general setting of polynomially compact operators.     

Uncomplementability of spaces of compact operators in larger spaces of operators

In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of c ₀ in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results in the paper.     

Two-weight Lp- Lq Bounds for Positive Dyadic Operators: Unified Approach to p ≤ q and p > q

We characterize the bounded and compact generalized Volterra companion integral operators acting between the standard Fock spaces. We also obtain asymptotic estimates for the norm of these operators in terms of certain Berezin type integral transforms which involve some function theoretic properties of symbols inducing the operators. As a special case, we prove that there exist no nontrivial compact Volterra companion integral and multiplication operators on Fock spaces.     

Weakly compact operators and multipliers on C0(S, A)

The problem of defining a convolution of operator-valued measures defined on a locally compact semitopological semigroup which represent weakly compact operators is considered in this paper. Convolution of the corresponding weakly compact operators is then defined. A characterization of multipliers on function spaces is given in terms of their representing measures.     

Compact AC(σ) Operators

All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) operator into real and imaginary parts.