Invariant subspaces of X** under the action of biconjugates

We study conditions on an infinite dimensional separable Banach space X implying that X is the only non-trivial invariant subspace of X** under the action of the algebra of biconjugates of bounded operators on . Such a space is called simple. We characterize simple spaces among spaces which contain an isomorphic copy of c ₀, and show in particular that any space which does not contain ℓ₁ and has property (u) of Pelczynski is simple.     

Some Algebraic Operators and the Invariant Subspace Problem

We consider the resolvent algebra ${R_A=\{T\in\mathcal{L} (X):\sup_{m \geq 0}\|(1+\,mA)T(1+\,mA)^{-1}\| < \infty \}}$ , and Deddens?? algebra ${B_A= \{T\in \mathcal{B}(H) : \sup_{n\geq 0}\|A^nTA^{-n}\|<\infty\}}$ . It is shown that both R _A and B _A?CI possess non-trivial invariant subspaces when A is an algebraic operator of degree 2. This assertion becomes stronger than the existence of a hyper-invariant subspace for R _A whenever R _A ?? {A}??. Investigation of the relationship between these two algebras is addressed for different classes of operators A. Also, a complete characterization of the algebra R _A when A is an algebraic operator is given. For the finite dimensional case, we present an elementary example showing that R _A contains properly {A}?? whenever A has an eigenvalue other than zero.     

Invariant and Hyperinvariant Subspaces of the Operator J α in Sobolev Spaces W p s [0,1]

Mathematical Notes -     

不变子空间的一个性质

给出了不变子空间的一个性质.应用此性质推广了文[1]中的定理1,并给出了几个推论.     

关于算子方程AX=XAX与AX=XA=XAX

用A的不变子空间作参数，给出了算子方程AX＝XAX的全部解。当A是单射或稠值域时，或者当A是正规算子时，给出了算子方程AX＝XA＝XAX的全部解。我们还给出正规算子X是算子方程AX＝XZ＝XAX的解的充分必要条件。     

Quasinilpotent Operators and the Invariant Subspace Problem

In this paper we construct quasinilpotent operators withoutnontrivial invariant subspaces. As well as being interestingin its own right, we believe that this is a sensible (perhapsnecessary) first step for addressing the difficult problem ofconstructing topologically simple Banach algebras.     

Invariant Subspaces for Positive Operators Acting on a Banach Space with Markushevich Basis

We introduce weak quasinilpotence for operators. Then, by substituting Markushevich basis and weak quasinilpotence at a nonzero vector for Schauder basis and quasinilpotence at a nonzero vector, respectively, we answer a question on the invariant subspaces of positive operators in [3].     

The Present State and Heritages of the Invariant Subspace Problem

No Abstract. . Received: February 2005     

Invariant Integrals for the Equilibrium Problem for a Plate with a Crack

We consider the equilibrium problem for a plate with a crack. The equilibrium of a plate is described by the biharmonic equation. Stress free boundary conditions are given on the crack faces. We introduce a perturbation of the domain in order to obtain an invariant Cherepanov–Rice-type integral which gives the energy release rate upon the quasistatic growth of a crack. We obtain a formula for the derivative of the energy functional with respect to the perturbation parameter which is useful in forecasting the development of a crack (for example, in study of local stability of a crack). The derivative of the energy functional is representable as an invariant integral along a sufficiently smooth closed contour. We construct some invariant integrals for the particular perturbations of a domain: translation of the whole cut and local translation along the cut.     

Invariant Differential Operators on Symmetric Cones and Hermitian Symmetric Spaces

We give a brief survey on the study of constructions of invariant differential operators on Riemannian symmetric spaces and of combinatorial and analytical properties of their eigenvalues, and pose some open questions.     

相应于一类对称自对偶李代数的顶点代数结构

设gsp是给定的对称自对偶李代数,给出了相应于gsp的一个顶点代数结构.     

Banach Lie algebras with Lie subalgebras of finite codimension: Their invariant subspaces and Lie ideals

The paper studies the existence of closed invariant subspaces for a Lie algebra L of bounded operators on an infinite-dimensional Banach space X. It is assumed that L contains a Lie subalgebra L₀ that has a non-trivial closed invariant subspace in X of finite codimension or dimension. It is proved that L itself has a non-trivial closed invariant subspace in the following two cases: (1) L₀ has finite codimension in L and there are Lie subalgebras L₀=L⁰⊂L¹⊂?⊂Lp=L such that Li+1=Li+[Li,Li+1] for all i; (2) L₀ is a Lie ideal of L and dim(L₀)=∞. These results are applied to the problem of the existence of non-trivial closed Lie ideals and closed characteristic Lie ideals in an infinite-dimensional Banach Lie algebra L that contains a non-trivial closed Lie subalgebra of finite codimension.     

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.     

On the Algebra of Pair Integral Operators with Homogeneous Kernels

In this paper, we study the Banach algebra generated by multidimensional pair integral operators with homogeneous kernels. We describe necessary and sufficient conditions for operators from the algebra to be Fredholm and present a formula for calculating the index.