On the maximal dual pairs of invariant subspaces of J-self-adjoint operators

In the J-spaces H=H₁ H₂, with the infinite-dimensional components H_k=P_kH (k = 1, 2), we can always find an operator A, for which there are at least two distinct invariant maximal dual pairs, such that if [x, x]=0 and [Ax, x]=0, then x=0.The author presented a suitable example at M. G. Krein's seminar in 1965.Translated from Mate matieheskie Zametki, Vol. 7, No. 4, pp. 443–447, April, 1970.     

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.     

On invariant subspaces of Volterra-type operators

The lattice of all the closed, invariant subspaces of the Volterra integration operator onL ²[0, 1] is equal to {B(a):a[0, 1]}, whereB(a)={fL ²[0, 1]:f=0 a.e. on [0,a]}. In order to extend this result to Banach function spaces we study the Volterra-type operatorV that was introduced in [7] for the case ofL ^p -spaces. Our main result characterizesL-closed subspaces of a Banach function spaceL that are invariant underV, whereL denotes the associate space ofL. In particular, if the norm ofL is order continuous and ifV is injective, then all the closed, invariant subspaces ofV are determined.This work was supported by the Research Ministry of Slovenia.     

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 invariant subspaces for polynomially bounded operators

We discuss the invariant subspace problem of polynomially bounded operators on a Banach space and obtain an invariant subspace theorem for polynomially bounded operators. At the same time, we state two open problems, which are relative propositions of this invariant subspace theorem. By means of the two relative propositions (if they are true), together with the result of this paper and the result of C.Ambrozie and V.Müller (2004) one can obtain an important conclusion that every polynomially bounded operator on a Banach space whose spectrum contains the unit circle has a nontrivial invariant closed subspace. This conclusion can generalize remarkably the famous result that every contraction on a Hilbert space whose spectrum contains the unit circle has a nontrivial invariant closed subspace (1988 and 1997).     

On quasispectral maximal subspaces of a class of Volterra-type operators

The concept of quasispectral maximal subspaces for quasinilpotent (but not nilpotent) operators was introduced by M. Omladi\v{c} in 1984. As an application a class of quasinilpotent operators on -spaces, close to the Volterra kernel operator, was studied. In the present Banach function space setting we determine all quasispectral maximal subspaces of analogues of such operators and prove that these subspaces are all the invariant bands. An example is given showing that (in general) they are not all the closed, invariant ideals of the operator.

     

Quasinormal operators on Krein space

In this paper we introduce the concept of quasinormal and subnormal operators on a Krein space and prove that every quasinormal operator is subnormal. And some conditions for an operator on a Hilbert space to be a subnormal operator in the Krein space sense are obtained.     

Compact perturbations of normal operators in a Krein space

We prove the existence of the spectral function of, in general, an unbounded normal operator in a Krein space that is the perturbation of a fundamentally reducible normal operator N by an operator of a special class that contain N-compact operators in it.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1299–1306, October, 1990.     

The lattices of invariant subspaces of a class of operators on the Hardy space

In the authors’ first paper, a Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. The current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator.     

Invariant subspaces of maximal J-dissipative operators

We prove a theorem on the existence of an invariant subspaces of special type for maximal J-dissipative and maximal J-symmetric operators.Translated from Matematicheskie Zametki, Vol. 12, No. 6, pp. 747–754, December, 1972.     

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.

     

Some invariant subspaces for subnormal operators

A theorem of D.E. Sarason is used to show that all subnormal operators have nontrivial invariant subspaces if some very special subnormal operators have them. It is then shown that these special subnormal operators as well as certain other operators do in fact have nontrivial invariant subspaces.     

On a class of selfadjoint operators in Krein space and their compact perturbations

For a class of selfadjoint operators in a Krein space containing the definitizable selfadjoint operators a funetional calculus and the spectral function are studied. Stability properties of the spectral function with respect to small compact perturbations of the resolvent are proved.     

Weak compactness in certain star-shift invariant subspaces

The context of much of the work in this paper is that of a backward-shift invariant subspace of the form , where B is some infinite Blaschke product. We address (but do not fully answer) the question: For which B can one find a (convergent) sequence in K_B such that the sequence of real measures converges weak-star to some nontrivial singular measure on ? We show that, in order for this to hold, K_B must contain functions with nontrivial singular inner factors. And in a rather special setting, we show that this is also sufficient. Much of the paper is devoted to finding conditions (on B) that guarantee that K_B has no functions with nontrivial singular inner factors. Our primary result in this direction is based on the “geometry” of the zero set of B.     

On polynomially bounded operators acting on a Banach space

By the Von Neumann inequality every contraction on a Hilbert space is polynomially bounded. A simple example shows that this result does not extend to Banach space contractions. In this paper, we give general conditions under which an arbitrary Banach space contraction is polynomially bounded. These conditions concern the thinness of the spectrum and the behaviour of the resolvent or the sequence of negative powers. To do this we use techniques from harmonic analysis, in particular, results concerning thin sets such as Helson sets, Kronecker sets and sets that satisfy spectral synthesis.