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A linear operator : between two topological vector spaces (TVSs) is called a bounded linear operator or just bounded if whenever is bounded in then () is bounded in . A subset of a TVS is called bounded (or more precisely, von Neumann bounded ) if every neighborhood of the origin absorbs it.
In its basic form, it asserts that for a family of continuous linear operators (and thus bounded operators) whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The theorem was first published in 1927 by Stefan Banach and Hugo Steinhaus, but it was also proven independently by Hans Hahn.
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Operators on these spaces are known as sequence transformations. Bounded linear operators over a Banach space form a Banach algebra in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of spectra that elegantly generalizes the theory of eigenspaces.
the operator norms of the operators T g are uniformly bounded. Then there is a positive invertible operator S on H such that S T g S −1 is unitary for every g in G. As a consequence, if T is an invertible operator with all its positive and negative powers uniformly bounded in operator norm, then T is conjugate by a positive invertible ...
The operator C can be defined by C(Bh) = Ah, extended by continuity to the closure of Ran(B), and by zero on the orthogonal complement of Ran(B). The operator C is well-defined since A*A ≤ B*B implies Ker(B) ⊂ Ker(A). The lemma then follows. In particular, if A*A = B*B, then C is a partial isometry, which is unique if Ker(B*) ⊂ Ker(C).
Bounded operator, a linear transformation L between normed vector spaces for which the ratio of the norm of L(v) to that of v is bounded by the same number over all non-zero vectors v. Unbounded operator, a linear operator defined on a subspace; Bounded poset, a partially ordered set that has both a greatest and a least element
Lemma — If A, B are bounded operators on a Hilbert space H, and A * A ≤ B * B, then there exists a contraction C such that A = CB. Furthermore, C is unique if ker( B * ) ⊂ ker( C ). The operator C can be defined by C ( Bh ) := Ah for all h in H , extended by continuity to the closure of Ran ( B ), and by zero on the orthogonal complement ...