Invariant subspaces for sequentially subdecomposable operators

In this paper, using Brown technique, we prove the Mohebi-Radjabalipour Conjecture by strengthening a slight thickness condition of the spectrum, and obtain some invariant subspace theorems. Our result contains an important known invariant subspace theorem as special cases.     

Invariant subspaces of positive strictly singular operators on Banach lattices

It is shown that every positive strictly singular operator T on a Banach lattice satisfying certain conditions is AM-compact and has invariant subspaces. Moreover, every positive operator commuting with T has an invariant subspace. It is also proved that on such spaces the product of a disjointly strictly singular and a regular AM-compact operator is strictly singular. Finally, we prove that on these spaces the known invariant subspace results for compact-friendly operators can be extended to strictly singular-friendly operators.     

On positive transitive operators

In this short paper we prove that even though the latest transitive operator has only one negative entry, we cannot get rid of it by simple operations. The research is motivated by, still unsolved, invariant subspace problem for positive operators.      

Beurling's theorem and invariant subspaces for the shift on Hardy spaces

Let G be a bounded open subset in the complex plane and let H~2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1-1 with respect to the Lebesgue measure on D and if the Riemann map belongs to the weak-star closure of the polynomials in H~∞(W). Our main theorem states: in order that for each M∈Lat (M_z), there exist u∈H~∞(G) such that M=∨{uH~2(G)}, it is necessary and sufficient that the following hold: (1) each component of G is a perfectly connected domain; (2) the harmonic measures of the components of G are mutually singular; (3) the set of polynomials is weak-star dense in H~∞(G). Moreover, if G satisfies these conditions, then every M∈Lat (M_z) is of the form uH~2(G), where u∈H~∞(G) and the restriction of u to each of the components of G is either an inner function or zero.     

Dissipative operators and additive perturbations in locally convex spaces

Let be a densely defined operator on a Banach space X. Characterizations of when generates a C₀‐semigroup on X are known. The famous result of Lumer and Phillips states that it is so if and only if is dissipative and is dense in X for some . There exists also a rich amount of Banach space results concerning perturbations of dissipative operators. In a recent paper Tyran–Kamińska provides perturbation criteria of dissipative operators in terms of ergodic properties. These results, and others, are shown to remain valid in the setting of general non–normable locally convex spaces. Applications of the results to concrete examples of operators on function spaces are also presented.     

Invariant subspaces for polynomially hyponormal operators

We show that if is a bounded operator on a Hilbert space such that for every polynomial , then has a nontrivial invariant subspace.

     

多圆盘的加权Bergman空间上的不变子空间和约化子空间

本文主要研究多圆盘的加权Bergman 空间上的不变子空间和约化子空间, 给出了某些解析Toeplitz 算子的极小约化子空间的完全刻画, 以及一类解析Toeplitz 算子Tzi (1≤i≤n) 的不变子空间的Beurling 型定理.     

Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions

We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C ₀-semigroup.     

On hyperinvariant subspaces of contraction operators on a Banach space whose spectrum contains the unit circle

In this paper, we prove that every operator in a class of contraction operators on a Banach space whose spectrum contains the unit circle has a nontrivial hyperinvariant subspace. This research is supported by the Natural Science Foundation of P. R. China (No. 10771039)     

On injective or dense-range operators leaving a given chain of subspaces invariant

In this paper we prove the existence of dense-range or one-to-one compact operators on a separable Banach space leaving a given finite chain of subspaces invariant. We use this result to prove that a semigroup of bounded operators is reducible if and only if there exists an appropriate one-to-one compact operator such that the collection of compact operators is reducible.

     

On principal invariant subspaces

Let F and G be closed subspaces of the complex Hilbert space H, and U and V be closed subspaces of F and G, respectively. In this paper, using the technique of operator block, we present the necessary and sufficient conditions under which (U, V) is a pair of (strictly, non-degenerate) principal invariant subspaces for (F, G).     

Hyperinvariant subspaces for quasinilpotent operators on Hilbert spaces

In this paper, we employ the model theory due to C. Foias and C. Pearcy and the notion of Enflo's extremal vectors of quasinilpotent operators to study the hyperinvariant subspace problem for quasinilpotent operators. Our main result is that if a quasinilpotent quasiaffinity T has a sequence of “c-eigenvectors” xn of TnT^∗n such that the set is compact, then T has a nontrivial hyperinvariant subspace.     

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.     

加权复合算子的不变测度

本文研究复的可分 Hilbert L2 ( S,Σ,μ)加权复合算子 Tf(· ) =w(· ) f( h(· ) )存在平方可积的不变测度的条件 ,并且证明关于 T不变的平方可积的测度支集全体所张成的闭子空间等于 T的幺模特征向量全体所张成的闭子空间     

Riesz transforms associated to Schrödinger operators on weighted Hardy spaces

Let w be some Ap weight and enjoy reverse Hölder inequality, and let L=−Δ+V be a Schrödinger operator on Rn, where is a non-negative function on Rn. In this article we introduce weighted Hardy spaces associated to L in terms of the area function characterization, and prove their atomic characters. We show that the Riesz transform ∇L−1/2 associated to L is bounded on for 1<p<2, and bounded from to the classical weighted Hardy space .     

Rank-one cross commutators on backward shift invariant subspaces on the bidisk

For a backward shift invariant subspace N in H^2（Г^2）, the operators Sz and Sw on N are defined by Sz = PNTz｜N and Sw, = PNTw｜N, where PN is the orthogonal projection from L^2（Г^2） onto N. We give a characterization of N satisfying rank [Sz, Sw^＊] = 1.     

Reducing subspaces of multiplication operators on function spaces

This survey presents the brief history and recent development on commutants and reducing subspaces of multiplication operators on both the Hardy space and the Bergman space, and von Neumann algebras generated by multiplication operators on the Bergman space.