Spectrum Reducing Extension for one Operator on a Banach Space TeX Export

@article{citeulike:4840187, abstract = {In this paper we show that, given an operator \$T\$ on a Banach space \$X\$, there is an extension \$Y\$ of \$X\$ such that \$T\$ extends in a natural way to an operator \$T^\sim\$ on \$Y\$, and the spectrum of \$T^\sim\$ is the approximate point spectrum of \$T\$. This answers a question posed by Bollobas, and contributes to a theory investigated by Shilov, Arens, Bollobas, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra \$L\$ containing elements \$g\_1\$ and \$g\_2\$ such that in no extension \$L'\$ of \$L\$ are the residual spectra of \$g\_1\$ and \$g\_2\$ eliminated simultaneously.}, author = {Read, C. J.}, citeulike-article-id = {4840187}, citeulike-linkout-0 = {http://dx.doi.org/10.2307/2000971}, citeulike-linkout-1 = {http://www.jstor.org/stable/2000971}, doi = {10.2307/2000971}, issn = {00029947}, journal = {Transactions of the American Mathematical Society}, keywords = {arens\_extn, cited, op-th}, number = {1}, pages = {413--429}, posted-at = {2009-06-14 01:34:36}, priority = {1}, publisher = {American Mathematical Society}, title = {Spectrum Reducing Extension for one Operator on a Banach Space}, url = {http://dx.doi.org/10.2307/2000971}, volume = {308}, year = {1988} }

TY - JOUR ID - citeulike:4840187 L3 - citeulike-article-id:4840187 N2 - In this paper we show that, given an operator $T$ on a Banach space $X$, there is an extension $Y$ of $X$ such that $T$ extends in a natural way to an operator $T^\sim$ on $Y$, and the spectrum of $T^\sim$ is the approximate point spectrum of $T$. This answers a question posed by Bollobas, and contributes to a theory investigated by Shilov, Arens, Bollobas, etc. The unusual transfinite construction is similar to that which we used earlier to find an inverse producing extension for a commutative unital Banach algebra which eliminates the residual spectrum of one element. We also give a counterexample, consisting of a Banach algebra $L$ containing elements $g_1$ and $g_2$ such that in no extension $L'$ of $L$ are the residual spectra of $g_1$ and $g_2$ eliminated simultaneously. IS - 1 JF - Transactions of the American Mathematical Society SN - 00029947 EP - 429 TI - Spectrum Reducing Extension for one Operator on a Banach Space PB - American Mathematical Society VL - 308 SP - 413 KW - arens_extn KW - cited KW - op-th AU - Read, CJ PY - 1988/// UR - http://dx.doi.org/10.2307/2000971 ER -