This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^p$-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^p$-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques...

2000 Mathematics Subject Classification: Primary 47B47, 47B10; Secondary 47A30.Let H be a separable infinite dimensional complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A ∈ L(H), the derivation δA : L(H)→ L(H) is defined by δA(X) = AX-XA. In this paper we prove that if A is an n-multicyclic hyponormal operator and T is hyponormal such that AT = TA, then || δA(X)+T|| ≥ ||T|| for all X ∈ L(H). We establish the same inequality if A is...

Let $X$ be a reflexive Banach space and $A$ be a closed operator in an $L_1$-space of $X$-valued functions. Then we characterize the range $R\left(A\right)$ of $A$ as follows. Let $0\ne {\lambda }_n\in \rho \left(A\right)$ for all $1\le n<\infty$, where $\rho \left(A\right)$ denotes the resolvent set of $A$, and assume that ${lim}_{n\to \infty }{\lambda }_n=0$ and ${sup}_{n\ge 1}\parallel {\lambda }_n{({\lambda }_n-A)}^{-1}\parallel <\infty$. Furthermore, assume that there exists ${\lambda }_{\infty }\in \rho \left(A\right)$ such that $\parallel {\lambda }_{\infty }{({\lambda }_{\infty }-A)}^{-1}\parallel \le 1$. Then $f\in R\left(A\right)$ is equivalent to ${sup}_{n\ge 1}{\parallel {({\lambda }_n-A)}^{-1}f\parallel }_1<\infty$. This generalizes Shaw’s result for scalar-valued functions.

Let $L\left(H\right)$ denote the algebra of operators on a complex infinite dimensional Hilbert space $H$. For $A,B\in L\left(H\right)$, the generalized derivation ${\delta }_{A,B}$ and the elementary operator ${\Delta }_{A,B}$ are defined by ${\delta }_{A,B}\left(X\right)=AX-XB$ and ${\Delta }_{A,B}\left(X\right)=AXB-X$ for all $X\in L\left(H\right)$. In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of ${\delta }_{A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of ${\Delta }_{A,B}$ with respect to the wider class of unitarily invariant norms on...

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.

An operator in a Banach space is called upper (lower) semi-Browder if it is upper (lower) semi-Fredholm and has a finite ascent (descent). We extend this notion to n-tuples of commuting operators and show that this notion defines a joint spectrum. Further we study relations between semi-Browder and (essentially) semiregular operators.