We show that for a linear space of operators the following assertions are equivalent. (i) is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map on a bilattice of subspaces determined by with and for any pair , and such that an operator lies in if and only if for all . This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.
We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson-Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function z on the vector-valued Lebesgue space L²(;ℂⁿ). Our approach allows us to prove an equivalent version of the vector-valued Helson-Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial isometries....
We present a change of variable method and use it to prove the equivalence to bundle shifts for certain analytic Toeplitz operators on the Banach spaces . In Section 2 we see this approach applied in the analysis of essential spectra. Some partial results were obtained in [9] in the Hilbert space case.
Using results on the reflexive algebra with two invariant subspaces, we calculate the hyperreflexivity constant for this algebra when the Hilbert space is two-dimensional. Then by the continuity of the angle for two subspaces, there exists a non-CSL hyperreflexive algebra with hyperreflexivity constant C for every C>1. This result leads to a kind of continuity for the hyperreflexivity constant.
An example of a nonzero quasinilpotent operator with reflexive commutant is presented.
A new example of a non-zero quasi-nilpotent operator T with reflexive commutant is presented. The norms converge to zero arbitrarily fast.
The uniqueness of the Wold decomposition of a finite-dimensional stationary process without assumption of full rank stationary process and the Lebesgue decomposition of its spectral measure is easily obtained.
The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been an important tool for studying such important problem. In this survey, we focus on Rota’s universal operators, pointing out their main properties and exhibiting some old and recent examples.
We construct a semigroup of bounded idempotents with no nontrivial invariant closed subspace. This answers a question which was open for some time.
This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read’s Banach space counter-example involving a finitely strictly singular operator.
The study of quasianalytic contractions, motivated by the hyperinvariant subspace problem, is continued. Special emphasis is put on the case when the contraction is asymptotically cyclic. New properties of the functional commutant are explored. Analytic contractions and bilateral weighted shifts are discussed as illuminating examples.