ON A CLASS OF GENERALIZED FREDHOLM OPERATORS, I

Christoph Schmoeger
<span title="1997-10-01">1997</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/jicn6vvuazg4bhlotllcfjfzsm" style="color: black;">Demonstratio Mathematica</a> </i> &nbsp;
Let X be an infinite-dimensional complex Banach space and let C(X) denote the Banach algebra of all bounded linear operators on X. We write $ g (X) for the following class of operators: TST = T and I -ST -TS is Fredholm}. Each Fredholm operator belongs to <$g(X). Operators in d> g (X) we call generalized Fredholm operators. In this paper we investigate the class <P g (X). Terminology and introduction Throughout this paper X denotes an infinite-dimensional complex Banach space and A denotes a
more &raquo; ... plex algebra with identity e 0 (the considerations in A will be purely algebraic). C(X) denotes the set of all bounded linear operators on X. For T G £(X) write N(T) and T(X) for the kernel and the range of T, respectively. T 6 C(X) is called Fredholm if dim N(T) and codim T(X) are both finite. We write $(X) for the set of all Fredholm operators on X. If T € $(X) then the index ind(T) of T is defined as An element t € A is called relatively regular if tst = t for some s € A. In this case we call s a pseudo-inverse of t. It is well known that Fredholm operators are relatively regular [4], § 74. Furthermore we have for T € £(X): T is relatively regular if and only if N(T) and T(X) are continuously projectable. 1991 Mathematics Subject Classification: 47A11.
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/dema-1997-0413">doi:10.1515/dema-1997-0413</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rf4oqnphnrgkdioxoxcmoavs2a">fatcat:rf4oqnphnrgkdioxoxcmoavs2a</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180727090510/https://www.degruyter.com/downloadpdf/j/dema.1997.30.issue-4/dema-1997-0413/dema-1997-0413.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/7f/30/7f30b5424a8a0bdbae2380c497d00962ba3918d1.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/dema-1997-0413"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> degruyter.com </button> </a>