In operator theory, a bounded operator T on a Banach space is said to be nilpotent if Tn = 0 for some positive integer n.[1] It is said to be quasinilpotent or topologically nilpotent if its spectrum σ(T) = {0}.
^Kreyszig, Erwin (1989). "Spectral Theory in Normed Spaces 7.5 Use of Complex Analysis in Spectral Theory, Problem 1. (Nilpotent operator)". Introductory Functional Analysis with Applications. Wiley. p. 393.
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In operator theory, a bounded operator T on a Banach space is said to be nilpotent if Tn = 0 for some positive integer n. It is said to be quasinilpotent...
an element x {\displaystyle x} of a ring R {\displaystyle R} is called nilpotent if there exists some positive integer n {\displaystyle n} , called the...
In linear algebra, a nilpotent matrix is a square matrix N such that N k = 0 {\displaystyle N^{k}=0\,} for some positive integer k {\displaystyle k} ...
V is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotentoperator. The operator norm of V is exactly ||V||...
be semisimple (resp. nilpotent) if ad ( x ) {\displaystyle \operatorname {ad} (x)} is a semisimple (resp. nilpotent) operator. If x ∈ g {\displaystyle...
nilpotent; but when raised to a square or higher power it gives itself as the result, it may be called idempotent. The defining equation of nilpotent...
the n-th iterate of T, Tn, is identically zero, then T is said to be nilpotent. If T2 = T, then T is said to be idempotent If T = kI, where k is some...
represents a class in the twisted cohomology with respect to the nilpotentoperator d + H {\displaystyle d+H} where d {\displaystyle d} is the ordinary...
{\displaystyle x:V\to V} as a sum of a semisimple endomorphism s and a nilpotent endomorphism n such that both s and n are polynomials in x. Jordan–Chevalley...
N, where D is diagonal and N is strictly upper triangular (and thus a nilpotent matrix). The diagonal matrix D contains the eigenvalues of A in arbitrary...
mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example...
separable Hilbert space H and λ(A) is an eigenvalue sequence. Every quasi-nilpotentoperator in a two-sided ideal satisfying (1) is a sum of commutators. A trace...
classification of Lie groups. Analogously to abelian, nilpotent, and solvable groups, one can define abelian, nilpotent, and solvable Lie algebras. A Lie algebra...
with M nilpotent, commuting with N and hence L. But then 0 = S2 − T2 = 2(L − M)(I + (L + M)/2). Since L and M commute, the matrix L + M is nilpotent and...
radical of a Noetherian ring is always nilpotent. The Levitzki radical is defined as the largest locally nilpotent ideal, analogous to the Hirsch–Plotkin...
The Jordan block corresponding to λ is of the form λI + N, where N is a nilpotent matrix defined as Nij = δi,j−1 (where δ is the Kronecker delta). The nilpotency...
Jordan–Chevalley decomposition expresses an operator as the sum of its semisimple (i.e., diagonalizable) part and its nilpotent part. Hence, a matrix is diagonalizable...
{\displaystyle \delta +d+\mathrm {more} } . s {\displaystyle s} is also nilpotent. s 2 = ( δ + d ) 2 = δ 2 + d 2 + ( δ d + d δ ) = 0 {\displaystyle s^{2}=(\delta...
with the group of unit quaternions. The Heisenberg group is a connected nilpotent Lie group of dimension 3 {\displaystyle 3} , playing a key role in quantum...
finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra...
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