A Hilbert space operator is called universal (in the sense of Rota) if every on similar to a multiple restriction universal one its invariant subspaces. We exhibit an analytic Toeplitz whose adjoint in Rota and which commutes with quasinilpotent, injective, compact dense range, but unlike other examples, it acts Bergman instead Hardy this associated “hyperbolic” composition ope...

A new example of a non-zero quasi-nilpotent operator T with reflexive commutant is presented. Norms ‖T n‖ converge to zero arbitrarily fast. Let H be a complex separable Hilbert space and let B(H) denote the algebra of all continuous linear operator on H. If T ∈ B(H) then {T}′ = {A ∈ B(H) : AT = TA} is called the commutant of T . By a subspace we always mean a closed linear subspace. If A ⊂ B(H...

In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only pair idempotents with commutator common subspace. We also present geometric characterization subspaces classify are essentially idempotent.

PREFACE 1. CONVERGENT. Does the set of cyclic operators have a non-empty interior?... 2. WEIGHTED. Is every part of a weighted shift similar to a weighted shift?. . . . 3. INVARIANT. If an intransitive operator has an inverse, is its inverse also intransitive? 4. TRIANGULAR. Is every normal operator the sum of a diagonal operator and a compact one? 5. DILATED. Is every subnormal Toeplitz operat...

“Weyl’s theorem” for an operator on a Hilbert space is a statement that the complement in the spectrum of the “Weyl spectrum” coincides with the isolated eigenvalues of finite multiplicity. In this paper we consider how Weyl’s theorem survives for polynomials of operators and under quasinilpotent or compact perturbations. First, we show that if T is reduced by each of its finite-dimensional eig...

We prove the existence of a non-trivial hyperinvariant subspace for several sets polynomially compact operators. The main results paper are: (i) norm closed algebra $$\mathcal {A}\subseteq \mathcal {B}(\mathscr {X})$$ which consists quasinilpotent operators has subspace; (ii) if there exists non-zero operator in closure generated by an band {S}$$ , then subspace.

For a bounded operator T acting on a complex Banach space, we show that if T −λ is not surjective, then λ is an isolated point of the surjective spectrum σsu(T ) of T if and only if X = H0(T−λ)+K(T−λ), where H0(T ) is the quasinilpotent part of T and K(T ) is the analytic core for T . Moreover, we study the operators for which dimK(T ) < ∞. We show that for each of these operators T , there exi...

In this paper we consider the question when an upward directed set of positive ideal-triangularizable operators on a Banach lattice is (simultaneously) ideal-triangularizable. We prove that a majorized upward directed set of ideal-triangularizable positive operators, which are compact or abstract integral operators is ideal-triangularizable. We also prove that a finite subset of an additive sem...

It is well known that if D is a bounded derivation on a Banach algebra A and if s is an element of A satisfying [s; Ds] = 0 then Ds must be quasinilpotent. The unbounded Kleinecke-Shirokov conjecture states that the same result holds even if D is unbounded. As yet, the conjecture has neither been proved nor has a case been found where it fails. If there is a counterexample we obtain a further r...