Open AccessTraces of operators and their history
Author(s) -
Albrecht Pietsch
Publication year - 2014
Publication title -
acta et commentationes universitatis tartuensis de mathematica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.276
H-Index - 6
eISSN - 2228-4699
pISSN - 1406-2283
DOI - 10.12697/acutm.2014.18.06
Subject(s) - nuclear operator , hilbert space , mathematics , trace (psycholinguistics) , compact operator on hilbert space , axiom , pure mathematics , operator (biology) , separable space , diagonal , operator theory , approximation property , property (philosophy) , banach space , rank (graph theory) , matrix (chemical analysis) , algebra over a field , operator norm , compact operator , mathematical analysis , computer science , combinatorics , extension (predicate logic) , materials science , repressor , linguistics , chemistry , composite material , biochemistry , geometry , epistemology , transcription factor , programming language , gene , philosophy
As well known, the trace of an n × n-matrix is dened to be the sum of all entries of the main diagonal. Extending this concept to the infinite-dimensional setting does not always work, since non-converging infinite series may occur. So one had to identify those operators that possess something like a trace. In a first step, integral operators generated from continuous kernels were treated. Then the case of operators on the infinite-dimensional separable Hilbert space followed. The situation in Banach spaces turned out to be more complicated, since the missing approximation property causes a lot of trouble. To overcome those difficulties, we present an axiomatic approach in which operator ideals play a dominant rule. The considerations include also singular traces that, by denition, vanish on all finite rank operators.
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