关于次可分解算子的不变子空间

In this paper,we prove the Mohebi-Radjabalipour Conjecture under an ad-ditional condition,and obtain an invariant subspace theorem on subdecomposableoperators.     

Shift-type invariant subspaces of contractions

Using the Sz.-Nagy-Foias functional model it was shown in [L. Kérchy, Injection of unilateral shifts into contractions with non-vanishing unitary asymptotes, Acta Sci. Math. (Szeged) 61 (1995) 443-476] that under certain conditions on a contraction T the natural embedding of a Hardy space of vector-valued functions into the corresponding L² space can be factored into the product of two transformations, intertwining T with a unilateral shift and with an absolutely continuous unitary operator, respectively. The norm estimates in the Factorization Theorem of this paper are sharpened to their best possible form by essential improvements in the proof. As a consequence we obtain that if the residual set of a contraction covers the whole unit circle then those invariant subspaces, where the restriction is similar to the unilateral shift with a similarity constant arbitrarily close to 1, span the whole space. Furthermore, the hyperinvariant subspace problem for asymptotically non-vanishing contractions is reduced to these special circumstances.     

Rank-one commutators on invariant subspaces of the Hardy space on the bidisk

We discuss on invariant subspaces of H²(Γ²) on which .     

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.     

On the existence of an invariant measure for Markov-Feller operators

Let X be a Polish space and P a Markov operator acting on the space of Borel measures on X. We will prove the existence of an invariant measure with respect to P, provided that P satisfies some condition of a Prokhorov type and that the family of functions is equi-continuous with respect to the Prokhorov distance at some point of the space X. Moreover, we will construct a counterexample which show that the above equi-continuity condition cannot be dropped.     

Hypergroups and invariant complemented subspaces

Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$, the class of left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$ and finally the class of non-zero left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$ in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$, and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$. By the help of these two characterizations, we extract some results about invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$.     

Spectra of some block operator matrices and application to transport operators

In this paper a new concept for a 3×3 block operator matrix is studied on a Banach space. It is shown that, under certain conditions, it defines a closable operator and its essential spectra are determined. Application to transport operators in L₁-space is given.     

Characterization of chaotic order and its applications to furuta's type operator inequalities

Very recently, we obtained a simple characterization of the chaotic order log A≥log B among positive invertible operators AB on a Hilbert space. In this note, we discuss Furuta's type operator inequalities as applications of our characterization of the chaotic order.     

Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators

Let H be a Hilbert space and let A be a simple symmetric operator in H with equal deficiency indices d:=n_±(A)<∞. We show that if, for all λ in an open interval I⊂R, the dimension of defect subspaces Nλ(A) (=Ker(A^?−λ)) coincides with d, then every self-adjoint extension has no continuous spectrum in I and the point spectrum of is nowhere dense in I. Application of this statement to differential operators makes it possible to generalize the known results by Weidmann to the case of an ordinary differential expression with both singular endpoints and arbitrary equal deficiency indices of the minimal operator.     

Dimension of measures invariant with respect to the Wa?ewska partial differential equation

We study asymptotic properties of a nonlinear first-order partial differential equation which describes the reproduction of blood cells. This equation under conditions proposed by Wa?ewska generates a semigroup of transformations with highly chaotic behaviour of trajectories. We show that this semigroup has invariant measures with arbitrary large dimension.     

Characterization of chaotic order and its application to furuta inequality

In this note, we give a simple characterization of the chaotic order among positive invertible operators on a Hilbert space. As an application, we discuss Furuta's type operator inequality.

     

On the Drazin inverse of the sum of two operators and its application to operator matrices

Given two bounded linear operators F,G on a Banach space X such that G²F=GF²=0, we derive an explicit expression for the Drazin inverse of F+G. For this purpose, firstly, we obtain a formula for the resolvent of an auxiliary operator matrix in the form . From the provided representation of D(F+G) several special cases are considered. In particular, we recover the case GF=0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207-217] for matrices and by Djordjevi? and Wei [D.S. Djordjevi?, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (1) (2002) 115-126] for operators. Finally, we apply our results to obtain representations for the Drazin inverse of operator matrices in the form which are extensions of some cases given in the literature.     

Fixed point theorems of block operator matrices on Banach algebras and an application to functional integral equations

In this paper, we study some fixed point theorems of a 2 × 2 block operator matrix defined on nonempty bounded closed convex subsets of Banach algebras, where the entries are nonlinear operators. Furthermore, we apply the obtained results to a coupled system of nonlinear equations. Copyright © 2012 John Wiley & Sons, Ltd.