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 $|T^n|$ converge to zero arbitrarily fast.

Let T be a spherical 2-expansive m-tuple and let $T^{}$ denote its spherical Cauchy dual. If $T^{}$ is commuting then the inequality $\genfrac{}{}{0pt}{}{{\sum }_{\left|\beta \right|=k}{(\beta !)}^{-1}{\left(T^{}\right)}^{\beta }\left(T^{}\right){*}^{\beta }\le (k+m-1}{k){\sum }_{\left|\beta \right|=k}{(\beta !)}^{-1}\left(T^{}\right){*}^{\beta }{\left(T^{}\right)}^{\beta }}$ holds for every positive integer k. In case m = 1, this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a 2-expansive (or concave) operator are hyponormal.

Let T: H → H be an operator in the complex Hilbert space H. Suppose that T is square bounded in average in the sense that there exists a constant M(T) with the property that, for all natural numbers n and for all x ∈ H, the inequality $1/(n+1){\sum }_{j=0}^n\parallel T^{j}x{\parallel }^2\le M{\left(T\right)}^2\parallel x{\parallel }^2$ is satisfied. Also suppose that the adjoint T* of the operator T is square bounded in average with constant M(T*). Then the operator T is power bounded in the sense that $sup\parallel T^{i}n\parallel :n\in \mathbb{N}$ is finite. In fact the following inequality is valid for all n ∈ ℕ: ∥Tn∥ ≤ e M(T)M(T*). Suppose...

In this paper we obtain some results concerning the set $ℳ=\cup \left\{\overline{R\left({\delta }_A\right)}\cap {\left\{A\right\}}^{\text{'}}A\in ℒ\left(H\right)\right\}$, where $\overline{R\left({\delta }_A\right)}$ is the closure in the norm topology of the range of the inner derivation ${\delta }_A$ defined by ${\delta }_A\left(X\right)=AX-XA.$ Here $\mathscr{H}$ stands for a Hilbert space and we prove that every compact operator in ${\overline{R\left({\delta }_A\right)}}^w\cap {\left\{A^{*}\right\}}^{\text{'}}$ is quasinilpotent if $A$ is dominant, where ${\overline{R\left({\delta }_A\right)}}^w$ is the closure of the range of ${\delta }_A$ in the weak topology.

Sea H un espacio de Hilbert complejo, separable y de dimensión infinita. Denotaremos por L(H) al álgebra de todos los operadores acotados en H. Carl Pearcy en 1977 introdujo el concepto de figura espectral de un operador T en L(H) [13]. Sin lugar a dudas hay dos resultados que hacen de la figura espectral de un operador un concepto importante. El primero se debe a Brown, Douglas y Fillmore:"Dos operadores esencialmente normales son débilmente equivalentes si y sólo si tienen la misma figura espectral".El...

2000 Mathematics Subject Classification: Primary 47A48, 93B28, 47A65; Secondary 34C94.New concepts of linear colligations and dynamic systems, corresponding to the linear operators, acting in the Banach spaces, are introduced. The main properties of the transfer function and its relation to the dual transfer function are established.