Let $N$ be a nilpotent operator over an infinite-dimensional (complex) Hilbert space $H$ with degree $\nu$. That is, $N^\nu = 0$ and $N^{\nu-1}\neq 0$. It is rather easy to show that $N^*$ is also nilpotent with degree at most $\nu-1$ since, for all $x,y\in H$, $$ 0 = \langle N^kx,y\rangle = \langle x, N^{k*}y\rangle. $$ But I would like to show that any composition of the form $$ N^j(N^{\nu-j})^* $$ or $$ (N^j)^*N^{\nu-j} $$ (where $j = 0, \dots, \nu$) is also nilpotent. I am not positive that this is true, but I can't see immediately why it is false either. Intuitively, I want to work with the relation between the kernel/range of $N$ and $N^*$ but have not made meaningful progress.
$\begingroup$
$\endgroup$
1
-
$\begingroup$ imho there is enough context. The question does not ask to prove the trivial claim ``$N^*$ is nilpotent'' but something else. $\endgroup$daw– daw2026-07-01 13:20:26 +00:00Commented 6 hours ago
Add a comment
|
1 Answer 1
$\begingroup$
$\endgroup$
1
As others have pointed out, if $\nu-k=k$ then $(N^{\nu-k})^*N^k$ is self-adjoint and cannot be zero.
Let $N$ be the $n\times n$-matrix $$ N = \pmatrix{ 0 & 1 \\ & \ddots & \ddots \\ &&\ddots&1 \\&&&0}, $$ and let $e_k$ denote the unit basis vectors. Take $k$ such that $n/2 < k < n$. Then the matrix $$ (N^{n-k})^T N^k $$ is a matrix that contains non-zero entries only at non-zero entries of $N^{2k-n}$, hence it is nilpotent. The same argument also applies to $1<k<n/2$.
Similarly, $(N^{n-k})^T N^k$ is nilpotent for nilpotent matrices $N$ such $S^{-1}NS = J$, where $J$ is the Jordan form of $N$ and $S$ is orthogonal.
For general $N$, the claim is no longer true, as the following random example (octave output) shows. It computes $M=S^{-1}NS$, with $N$ a nice nilpotent matrix as above, random $S$ (i.e. not orthogonal). Sets $M:= S^{-1}NS$ nilpotent, and computes $T=M^T M^3$ and $T^4\ne0$.
octave:47> N
N =
0 1 0 0
0 0 1 0
0 0 0 1
0 0 0 0
octave:48> S = rand(4)
S =
0.098808 0.386082 0.984632 0.513226
0.920632 0.870510 0.248718 0.495748
0.584109 0.061110 0.491139 0.090677
0.199199 0.203628 0.599496 0.061701
octave:49> M = S^(-1)*N*S
M =
0.062923 -0.038031 0.993985 -0.096602
-0.737021 -1.225097 -1.216917 -0.507592
-0.013713 0.197602 -0.051503 0.079597
2.362441 2.245968 1.307504 1.213678
octave:50> T = M'*M*M*M
T =
1.2053 1.2321 3.6275 0.3734
1.2575 1.2855 3.7846 0.3895
0.7251 0.7412 2.1822 0.2246
0.6575 0.6721 1.9787 0.2036
octave:51> T^4
ans =
139.793 142.902 420.713 43.300
145.845 149.089 438.927 45.175
84.095 85.965 253.087 26.048
76.251 77.947 229.481 23.619
-
$\begingroup$ Ah, the self-adjoint notion should have been obvious. Thank you! $\endgroup$saas17– saas172026-07-01 16:05:56 +00:00Commented 4 hours ago