In this paper we consider operators acting on a subspace $ℳ$ of the space $L_2({ℝ}^m;{ℂ}_m)$ of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace $ℳ$ is defined as the orthogonal sum of spaces ${ℳ}_{s,k}$ of specific Clifford basis functions of $L_2({ℝ}^m;{ℂ}_m)$. Every Clifford endomorphism of $ℳ$ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they...

We establish interpolation properties under limiting real methods for a class of closed ideals including weakly compact operators, Banach-Saks operators, Rosenthal operators and Asplund operators. We show that they behave much better than compact operators.

Conditions involving closed range of multipliers on general Banach algebras are studied. Numerous conditions equivalent to a splitting A = TA ⊕ kerT are listed, for a multiplier T defined on the Banach algebra A. For instance, it is shown that TA ⊕ kerT = A if and only if there is a commuting operator S for which T = TST and S = STS, that this is the case if and only if such S may be taken to be a multiplier, and that these conditions are also equivalent to the existence of a factorization T = PB,...

Let denote a complex, infinite-dimensional, separable Hilbert space, and for any such Hilbert space , let () denote the algebra of bounded linear operators on . We show that for any co-analytic, right-invertible T in (), αT is hypercyclic for every complex α with $\left|\alpha \right|>{\beta }^{-1}$, where $\beta \equiv in{f}_{\left|\right|x\left|\right|=1}\left|\right|T*x\left|\right|>0$. In particular, every co-analytic, right-invertible T in () is supercyclic.

The main meaning of the common extension for two linear operators is the following: given two vector subspaces G₁ and G₂ in a vector space (respectively an ordered vector space) E, a Dedekind complete ordered vector space F and two (positive) linear operators T₁: G₁ → F, T₂: G₂ → F, when does a (positive) linear common extension L of T₁, T₂ exist? First, L will be defined on span(G₁ ∪ G₂). In other results, formulated in the line of the Hahn-Banach extension theorem, the common...

We study a certain operator of multiplication by monomials in the weighted Bergman space both in the unit disk of the complex plane and in the polydisk of the $n$-dimensional complex plane. Characterization of the commutant of such operators is given.

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.

This paper characterizes the commutant of certain multiplication operators on Hilbert spaces of analytic functions. Let $A=M_z$ be the operator of multiplication by z on the underlying Hilbert space. We give sufficient conditions for an operator essentially commuting with A and commuting with $A^n$ for some n>1 to be the operator of multiplication by an analytic symbol. This extends a result of Shields and Wallen.

MSC 2010: Primary: 447B37; Secondary: 47B38, 47A15