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.     

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 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).     

Nearly invariant subspaces for shift semigroups

Let{T (t)}_t≥0 be a C₀-semigroup on an infinite-dimensional separable Hilbert space; a suitable definition of near{T (t)~*}_t≥0 invariance of a subspace is presented in this paper. A series of prototypical examples for minimal nearly{S(t)~*}t≥0 invariant subspaces for the shift semigroup{S(t)}_t≥0 on L²(0,∞) are demonstrated, which have close links with near T~*_θ invariance on Hardy spaces of the unit disk for an inner function θ. Especially, ...     

On semigroups generated by differential operators on Lie groups

Let dR be the differential of a strongly continuous representation of a Lie group G on a Hilbert space H. Let P be the left-invariant second-order differential operator on G with positive semidefinite main part P₂ and with first-order part P₁. If Im(P₁) is in some sense subordinate to P₂ then dR(P) is a pregenerator of a strongly continuous semigroup of operators in H. If the whole P₁ is in some sense subordinate to P₂ then that semigroup is holomorphic or, even more, dR(P) is a pregenerator of a cosine operator function.     

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.     

Generation of analytic semigroups by degenerate elliptic operators

We consider the generation of analytic semigroups by degenerate elliptic operators satisfying a uniform H?rmander condition under homogeneous Dirichlet boundary conditions. Received January 12, 1996     

Common invariant subspaces for collections of operators

Let be a collection of bounded operators on a Banach spaceX of dimension at least two. We say that is finitely quasinilpotent at a vectorx ₀X whenever for any finite subset of the joint spectral radius of atx ₀ is equal 0. If such collection contains a non-zero compact operator, then and its commutant have a common non-trivial invariant, subspace. If in addition, is a collection of positive operators on a Banach lattice, then has a common non-trivial closed ideal. This result and a recent remarkable theorem of Turovskii imply the following extension of the famous result of de Pagter to semigroups. Let be a multiplicative semigroup of quasinilpotent compact positive operators on a Banach lattice of dimension at least two. Then has a common non-trivial invariant closed ideal.This work was supported by the Research Ministry of Slovenia.     

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.     

Stability of semigroups of operators and spectral subspaces

No abstract. February 17, 1998     

Some characterizations of Riesz operators by means of invariant subspaces

In this paper we shall give several characterizations of Riesz operatorsT on Banach spaces by means of some closedT-invariant subspaces. Moreover we also give a characterization of these operators dual to that give by Dieudonné in [5].     

Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

Semigroup Forum - We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$mathrm {C}(partial M)$$ of continuous functions on the boundary...     

Quasi-L p -contractive analytic semigroups generated by elliptic operators with complex unbounded coefficients on arbitrary domains

Let ?? be a domain in ? ^N and consider a second order linear partial differential operator A in divergence form on ?? which is not required to be uniformly elliptic and whose coefficients are allowed to be complex, unbounded and measurable. Under rather general conditions on the growth of the coefficients we construct a quasi-contractive analytic semigroup $(e^{-t A_{V}})_{t\geqslant0}$ on L ²(??,dx), whose generator A _V gives an operator realization of A under general boundary conditions. Under suitable additional conditions on the imaginary parts of the diffusion coefficients, we prove that for a wide class of boundary conditions, the semigroup $(e^{-t A_{V}})_{t\geqslant0}$ is quasi-L ^p -contractive for 1<p<??. Similar results hold for second order nondivergence form operators whose coefficients satisfy conditions similar to those on the coefficients of the operator A, except for some further requirements on the diffusion coefficients. Some examples where our results can be applied are provided.