In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm of a linear map is the maximum factor by which it "lengthens" vectors.
the operatornorm measures the "size" of certain linear operators by assigning each a real number called its operatornorm. Formally, it is a norm defined...
standard basis, and one defines the corresponding induced norm or operatornorm or subordinate norm on the space K m × n {\displaystyle K^{m\times n}} of...
M} is called the operatornorm of L {\displaystyle L} and denoted by ‖ L ‖ . {\displaystyle \|L\|.} A bounded operator between normed spaces is continuous...
Matrix norm, a map that assigns a length or size to a matrix Operatornorm, a map that assigns a length or size to any operator in a function space Norm (abelian...
Schatten norm (or Schatten–von-Neumann norm) arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm. Let...
logarithmic norm is a real-valued functional on operators, and is derived from either an inner product, a vector norm, or its induced operatornorm. The logarithmic...
one can say that the weak-operator and σ-weak topologies agree on norm-bounded sets in B(H): Every trace-class operator is of the form S = ∑ i λ i u...
mathematics, a compact operator is a linear operator T : X → Y {\displaystyle T:X\to Y} , where X , Y {\displaystyle X,Y} are normed vector spaces, with...
Theorems 1 and 2 below.) The dual norm is a special case of the operatornorm defined for each (bounded) linear map between normed vector spaces. Since the ground...
reference to algebras of operators on a separable Hilbert space, endowed with the operatornorm topology. In the case of operators on a Hilbert space, the...
operator 2-norm. One can easily verify the relationship between the Ky Fan 1-norm and singular values. It is true in general, for a bounded operator M...
operatornorm? Every finite-dimensional reflexive algebra is hyper-reflexive. However, there are examples of infinite-dimensional reflexive operator algebras...
linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces...
space X {\displaystyle X} . If the Neumann series converges in the operatornorm, then Id − T {\displaystyle {\text{Id}}-T} is invertible and its inverse...
transpose, of an operator A : E → F {\displaystyle A:E\to F} , where E , F {\displaystyle E,F} are Banach spaces with corresponding norms ‖ ⋅ ‖ E , ‖ ⋅ ‖...
some operator T on X. This could have several different meanings: If ‖ T n − T ‖ → 0 {\displaystyle \|T_{n}-T\|\to 0} , that is, the operatornorm of T...
. In this case, its operatornorm is equal to ‖ f ‖ ∞ {\displaystyle \|f\|_{\infty }} . The adjoint of a multiplication operator T f {\displaystyle T_{f}}...
set of Fredholm operators from X to Y is open in the Banach space L(X, Y) of bounded linear operators, equipped with the operatornorm, and the index is...
linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operatornorm. The...
spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue...
formula, also holds for bounded linear operators: letting ‖ ⋅ ‖ {\displaystyle \|\cdot \|} denote the operatornorm, we have ρ ( A ) = lim k → ∞ ‖ A k ‖...
is a quasinilpotent operator (that is, the spectral radius, ρ(V), is zero), but it is not nilpotent operator. The operatornorm of V is exactly ||V||...
Conversely, if an operator is bounded, then it is continuous. The space of such bounded linear operators has a norm, the operatornorm given by ‖ A ‖ =...
closure of finite-rank operators (representable by finite-dimensional matrices) in the topology induced by the operatornorm. As such, results from matrix...
(σ1(T), σ2(T), …). The largest singular value σ1(T) is equal to the operatornorm of T (see Min-max theorem). If T acts on Euclidean space R n {\displaystyle...