Suppose that M is a lower triangular infinite matrix with constant row segments given by the sequence
\begin{equation}
ln (1 + 1/(n + 1)).
\end{equation}
Then the self-commutator of M is a positive operator on the Hilbert space of square summable sequences. This can be verified using the Maclaurin series expansion for ln(1+x) together with the main result from "Hyponormal terraced matrices," Far East J. Math. Sci. 5 (1997), no. 3, 425--428.
Note: This approach does not work for sin(1/(n+1)) or Arctan(1/(n+1)), since they do not satisfy the upper inequality in the reference cited.
An affirmative result can also be obtained for tan(1/(n+2)) using Maclaurin series expansions for sin x and cos x. Note: tan 1/(n+1)) was not used because it does not satisfy the initial condition in the reference cited.
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ReplyDeleteComputer software estimates seem to suggest a conjecture that Arcsin(1/(n+2)) and sinh(1/(n+2)) also satisfy the conditions in the reference cited.
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