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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Similarity orbits and the range of the generalized derivation $X\to MX-XN$
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by Allen Schweinsberg PDF
Trans. Amer. Math. Soc. 324 (1991), 201-211 Request permission

Abstract:

If $M$ and $N$ are bounded operators on infinite dimensional complex Hilbert spaces $\mathcal {H}$ and $\mathcal {K}$, let $\tau (X) = MX - XN$ for $X$ in $\mathcal {L}(\mathcal {K},\mathcal {H})$. The closure of the range of $\tau$ is characterized when $M$ and $N$ are normal. There is a close connection between the range of $\tau$ and operators $C$ for which $[\begin {array}{*{20}{c}} M \& C \\ 0 \& N \\ \end {array} ]$ is in the closure of the similarity orbit of $[\begin {array}{*{20}{c}} M \& 0 \\ 0 ^ N \\ \end {array} ]$. This latter set is characterized and compared with the closure of the range of $\tau$.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 324 (1991), 201-211
  • MSC: Primary 47A65; Secondary 47B15, 47B47
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1005938-1
  • MathSciNet review: 1005938