We prove that a function f is a polynomial if G◦f is a polynomial for every bounded linear functional G. We also show that an operator-valued function is a polynomial if it is locally a polynomial.

In addition to Pisier’s counterexample of a non-accessible maximal Banach ideal, we will give a large class of maximal Banach ideals which are accessible. The first step is implied by the observation that a “good behaviour” of trace duality, which is canonically induced by conjugate operator ideals can be extended to adjoint Banach ideals, if and only if these adjoint ideals satisfy an accessibility condition (theorem 3.1). This observation leads in a natural way to a characterization of accessible...

Using elementary arguments we improve former results of P. Vrbová concerning local spectrum. As a consequence, we obtain a new proof of Kaplansky’s theorem on algebraic operators on a Banach space.

Let be a complex Banach space and e ∈ a nonzero vector. Then the set of all operators T ∈ ℒ() with ${\sigma }_T\left(e\right)={\sigma }_{\delta }\left(T\right)$, respectively $r_T\left(e\right)=r\left(T\right)$, is residual. This is an analogy to the well known result for a fixed operator and variable vector. The results are then used to characterize linear mappings preserving the local spectrum (or local spectral radius) at a fixed vector e.

Let $ℒ\left(\mathscr{H}\right)$ be the algebra of all bounded linear operators on a complex Hilbert space $\mathscr{H}$. We characterize locally spectrally bounded linear maps from $ℒ\left(\mathscr{H}\right)$ onto itself. As a consequence, we describe linear maps from $ℒ\left(\mathscr{H}\right)$ onto itself that compress the local spectrum.

Concise but self-contained reviews are given on theories of majorization and symmetrically normed ideals, including the proofs of the Lidskii-Wielandt and the Gelfand-Naimark theorems. Based on these reviews, we discuss logarithmic majorizations and norm inequalities of Golden-Thompson type and its complementary type for exponential operators on a Hilbert space. Furthermore, we obtain norm convergences for the exponential product formula as well as for that involving operator means.

Let $A={\left(a_{n,k}\right)}_{n,k\ge 1}$ be a non-negative matrix. Denote by $L_{v,p,q,F}\left(A\right)$ the supremum of those $L$ that satisfy the inequality $${\parallel Ax\parallel }_{v,q,F}\ge L{\parallel x\parallel }_{v,p,F},$$ where $x\ge 0$ and $x\in l_p(v,F)$ and also $v={\left(v_n\right)}_{n=1}^{\infty }$ is an increasing, non-negative sequence of real numbers. If $p=q$, we use $L_{v,p,F}\left(A\right)$ instead of $L_{v,p,p,F}\left(A\right)$. In this paper we obtain a Hardy type formula for $L_{v,p,q,F}\left(H_{\mu }\right)$, where $H_{\mu }$ is a Hausdorff matrix and $0<q\le p\le 1$. Another purpose of this paper is to establish a lower bound for $\parallel A_W^{NM}{\parallel }_{v,p,F}$, where $A_W^{NM}$ is the Nörlund matrix associated with the sequence $W={\left\{w_n\right\}}_{n=1}^{\infty }$ and $1<p<\infty$. Our results generalize some works of Bennett, Jameson and present authors....