부산대학교 도서관

Let $\scr{X}=(x_{j})_{j\in J}$ be a family of vectors in a given Banach space. A bounded linear operator $T$ on the linear span of $\scr{X}$ is said to be an $\scr{X}${\it -multiplier} if there exist scalars $\lambda _{j}(T)$ such that $Tx_{j}=\lambda _{j}(T)x_{j}$ for all $j\in J.$ The family $\scr{X}$ is said to have the {\it spectral localization property} (SLP) if, for each $\scr{X}$-multiplier $T,$ the spectrum of $T $ coincides with the closure of the set $\{\lambda _{j}(T)\colon j\in J\}.$ It is known that every unconditional basis has the SLP, but this is not the case for arbitrary Schauder bases. \par Among other things, the article under review provides the construction of a Muckenhoupt exponential basis without the SLP in the space $L^{2}(\Bbb{T},w),$ where $\Bbb{T}$ denotes the unit circle and $w$ is a suitable weight function on $\Bbb{T}.$ It is also shown that, for every Schauder basis $\scr{X}$ of a Hilbert space, the SLP implies that, for each $\delta >0,$ there exists some constant $c>0$ such that $\left\Vert T^{-1}\right\Vert \leq c$ for every $\scr{X}$-multiplier $T$ for which $\left\Vert T\right\Vert \leq 1$ and $\left\vert \lambda _{j}(T)\right\vert \geq \delta $ for all $j\in J.$ An example is given to show that this result does not extend to arbitrary Banach spaces. The article concludes with a brief discussion of the spectral localization property for $C_{0}$-semigroups.