Invariant subspaces for polynomially bounded operators

Let T be a polynomially bounded operator on a Banach space X whose spectrum contains the unit circle. Then T∗ has a nontrivial invariant subspace. In particular, if X is reflexive, then T itself has a nontrivial invariant subspace. This generalizes the well-known result of Brown, Chevreau, and Pearcy for Hilbert space contractions.     

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 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.     

Laplace transforms of polynomially bounded vector-valued functions and semigroups of operators

We define an invariant of measure-theoretic isomorphism for dynamical systems, as the growth rate inn of the number of small -balls aroundα-n-names necessary to cover most of the system, for any generating partitionα. We show that this rate is essentially bounded if and only if the system is a translation of a compact group, and compute it for several classes of systems of entropy zero, thus getting examples of growth rates inO(n),O(n ^k ) fork ε ℕ, oro(f(n)) for any given unboundedf, and of various relationships with the usual notion of language complexity of the underlying topological system.     

On the similarity problem for polynomially bounded operators on hilbert space

Partial solutions are obtained to Halmos’ problem, whether or not any polynomially bounded operator on a Hilbert spaceH is similar to a contraction. Central use is made of Paulsen’s necessary and sufficient condition, which permits one to obtain bounds on ‖S‖ ‖S ^?1‖, whereS is the similarity. A natural example of a polynomially bounded operator appears in the theory of Hankel matrices, defining $$R_f = \left( {\begin{array}{*{20}c} {S*} \\ 0 \\ \end{array} \begin{array}{*{20}c} {\Gamma _f } \\ S \\ \end{array} } \right)$$ onl ² ⊕l ², whereS is the shift and Γ _f the Hankel operator determined byf withf′ ∈ BMOA. Using Paulsen’s condition, we prove thatR _f is similar to a contraction. In the general case, combining Grothendieck’s theorem and techniques from complex function theory, we are able to get in the finite dimensional case the estimate $$\left\| S \right\|\left\| {S^{ - 1} } \right\| \leqq M^4 log(dim H)$$ whereSTS ^?1 is a contraction and assuming \(\left\| {p\left( T \right)} \right\| \leqq M\left\| p \right\|_\infty \) wheneverp is an analytic polynomial on the disc.     

Locally polynomially bounded structures

We prove a theorem which provides a method for constructingpoints on varieties defined by certain smooth functions. Werequire that the functions be definable in a definably completeexpansion of a real closed field and be locally definable ina fixed o-minimal and polynomially bounded reduct. As an applicationwe show that in certain o-minimal structures, definable functionsare piecewise implicitly defined over the basic functions inthe language.     

On algebras of polynomially compact operators

In this paper, we find sufficient and necessary conditions for a triangularizable closed algebra of polynomially compact operators to be commutative modulo the radical. We also prove that an algebraic algebra of operators of a bounded degree on a Banach space is triangularizable under some mild additional conditions. As a special case we obtain a result stating that every algebraic algebra of operators of bounded degree is triangularizable whenever its commutators are nilpotent operators.     

Differential equations over polynomially bounded o-minimal structures

We investigate the asymptotic behavior at of non-oscillatory solutions to differential equations a$">, where is definable in a polynomially bounded o-minimal structure. In particular, we show that the Pfaffian closure of a polynomially bounded o-minimal structure on the real field is levelled.

     

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.

     

Asymptotic behavior of one-parameter semigroups of positive operators

In this paper we survey the Perron-Frobenius spectral theory for positive semigroups on Banach lattices and indicate its applications to stability theory of retarded differential equations and quasi-periodic flows.     

A functional model for polynomially posinormal operators

In this paper we introduce a class of operators naturally extending the classes of hyponormal and posinormal operators. For this class we construct a generating family of eigendistributions, unitary invariants and a functional model.     

概率型算子在Lp空间的渐近性质

In 1985, Khan, R. A. established the asymptotic formulas of operators of probabilistic type inL1, space by introducing a newLp-norm. The purpose of this paper is to study the asymptotic rate of these operators, inLp (p>1) spaces. Project supported by the National Natural Science Foundation of China     

Infinitely Peano differentiable functions in polynomially bounded o-minimal structures

Let R be an o-minimal expansion of a real closed field. We show that the definable infinitely Peano differentiable functions are smooth if and only if R is polynomially bounded.