On a class of -contractions: hyperinvariant subspaces and intertwining operators.
On a decomposition for pairs of commuting contractions
A new decomposition of a pair of commuting, but not necessarily doubly commuting contractions is proposed. In the case of power partial isometries a more detailed decomposition is given.
On BMO-type characteristics for one class of holomorphic functions in a disk.
On class A operators
We show that every class A operator has a scalar extension. In particular, such operators with rich spectra have nontrivial invariant subspaces. Also we give some spectral properties of the scalar extension of a class A operator. Finally, we show that every class A operator is nonhypertransitive.
On commuting isometries
On decomposition of pairs of commuting isometries
A review of known decompositions of pairs of isometries is given. A new, finer decomposition and its properties are presented.
On Deddens΄s Theorem
On Dominant Operators.
On existence of hyperinvariant subspaces for linear maps.
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...
On operator bands
A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on that is weakly...
On pre-reflexive algebras and pre-reflexive operators
On reflexive subobject lattices and reflexive endomorphism algebras
In this paper we study the reflexive subobject lattices and reflexive endomorphism algebras in a concrete category. For the category Set of sets and mappings, a complete characterization for both reflexive subobject lattices and reflexive endomorphism algebras is obtained. Some partial results are also proved for the category of abelian groups.
On reflexivity and hyperreflexivity of some spaces of intertwining operators
Let be weak contractions (in the sense of Sz.-Nagy and Foiaş), the minimal functions of their parts and let be the greatest common inner divisor of . It is proved that the space of all operators intertwining is reflexive if and only if the model operator is reflexive. Here means the compression of the unilateral shift onto the space . In particular, in finite-dimensional spaces the space is reflexive if and only if all roots of the greatest common divisor of minimal polynomials...
On strong generation of B(ℋ) by two commutative C*-algebras
The algebra B(ℋ) of all bounded operators on a Hilbert space ℋ is generated in the strong operator topology by a single one-dimensional projection and a family of commuting unitary operators with cardinality not exceeding dim ℋ. This answers Problem 8 posed by W. Żelazko in [6].
On the equations X=KXS and AX=XK
On the invariant subspace problem in Banach spaces
On the orbit of the centralizer of a matrix
Let A be a complex n × n matrix. Let A' be its commutant in Mₙ(ℂ), and C(A) be its centralizer in GL(n,ℂ). Consider the standard C(A)-action on ℂⁿ. We describe the C(A)-orbits via invariant subspaces of A'. For example, we count the number of C(A)-orbits as well as that of invariant subspaces of A'.
On the reflexivity and hyperreflexivity of algebras and subspaces
A review of recent reflexivity and hyperreflexivity results is presented. We concentrate particularly on a finite-dimensional situation, Toeplitz operators and partial isometries. Open problems in this area are given.